https://michaelnielsen.org/polymath/api.php?action=feedcontributions&feedformat=atom&user=Klasm
Polymath Wiki - User contributions [en]
2026-04-21T05:10:26Z
User contributions
MediaWiki 1.42.3
https://michaelnielsen.org/polymath/index.php?title=%22Low_Dimensions%22_grant_acknowledgments&diff=3083
"Low Dimensions" grant acknowledgments
2010-04-25T19:54:42Z
<p>Klasm: </p>
<hr />
<div>Participants should be arranged in alphabetical order of surname.<br />
<br />
== Participants and contact information ==<br />
<br />
* Kevin O'Bryant, CUNY (Staten Island and the Graduate Center), [http://www.math.csi.cuny.edu/obryant]<br />
* Klas Markström, Umeå universitet, Sweden. [http://abel.math.umu.se/~klasm/]<br />
* Terence Tao, UCLA, [http://www.math.ucla.edu/~tao]<br />
<br />
== Grant information ==<br />
<br />
* Kevin O'Bryant is supported by a grant from The City University of New York PSC-CUNY Research Award Program.<br />
* Terence Tao is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2968
Omega(N)
2010-02-02T13:11:10Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}.</math><br />
<br />
A variant of the above expression for <math>\omega(N)</math> is<br />
<br />
:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math><br />
<br />
<br />
== Exact values ==<br />
<br />
These are the currently known values of <math>\Omega(N)</math><br />
<br />
<math>\Omega(1)=1</math> at <math>y_1=0</math><br />
<br />
<math>\Omega(2)=\frac{1}{2}</math> at <math>y_1=0, y_2=-\infty</math><br />
<br />
<math>\Omega(3)=\frac{1}{2}</math> at <math>y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}}</math><br />
<br />
<math>\Omega(4)=\frac{1}{2}</math> at <math>y_1=0, y_2=\infty, y_3=\infty</math><br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,84</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
A plot of the bounds for <math> N=1,...,84</math><br />
[http://abel.math.umu.se/~klasm/Data/EDP/omega-bound-plot.pdf]<br />
<br />
<nowiki><br />
<br />
</nowiki><br />
<br />
{<br />
{1, <br />
{y[1]->0}},<br />
{1/2, <br />
{y[2] -> -Infinity}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -Infinity, y[3] -> Infinity}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, <br />
{y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, <br />
{y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*10^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, <br />
{y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, <br />
{y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, <br />
{y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, <br />
{y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, <br />
{y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*10^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*10^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*10^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*10^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*10^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*10^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*10^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*10^6, <br />
y[47] -> 1.3757810922960734*10^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*10^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*10^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*10^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*10^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, <br />
{y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*10^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*10^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*10^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*10^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*10^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*10^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, <br />
{y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*10^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, <br />
{y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*10^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*10^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*10^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*10^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*10^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*10^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*10^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*10^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, <br />
{y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*10^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, <br />
{y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*10^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*10^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*10^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*10^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*10^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, <br />
{y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*10^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*10^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*10^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, <br />
{y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*10^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*10^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*10^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*10^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*10^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}},<br />
{0.5347953853151964, <br />
{y[2] -> 826616.6264389796, <br />
y[3] -> -3.2445194137959525, y[5] -> 1.1337838988884934*10^6, <br />
y[7] -> -0.053183445803101505, y[11] -> 813021.9382822418, <br />
y[13] -> -0.35844167042277564, y[17] -> 5.339307410851948, <br />
y[19] -> 0.219050505622947, y[23] -> 499949.5722852526, <br />
y[29] -> -603285.5606499284, y[31] -> 0.2483912347244814, <br />
y[37] -> -0.5099629142111536, y[41] -> 1.0913022635794224*10^6, <br />
y[43] -> 0.16012687574803594, y[47] -> 1.4162187543877018, <br />
y[53] -> -708771.0434949356, y[59] -> 4.926882173473444, <br />
y[61] -> -0.8700222063716009, y[67] -> -0.29467048865194995, <br />
y[71] -> 294424.5531175645, y[73] -> 0.018929724818790454}}, <br />
{0.5475609984129808, <br />
{y[2] -> -25.165389880489784, <br />
y[3] -> -2.3579309267835553, y[5] -> -8.785814809607855, <br />
y[7] -> 0.04671421855986395, y[11] -> -3.491265944223444, <br />
y[13] -> -0.04348811944806861, y[17] -> 10656.226669356573, <br />
y[19] -> 0.4981709858168526, y[23] -> -2.958123944570159, <br />
y[29] -> -12.528475056708524, y[31] -> 0.3726267566768777, <br />
y[37] -> -0.34033918120195966, y[41] -> 0.840340004423624, <br />
y[43] -> -3.3868866214535616, y[47] -> 1.675744684738037, <br />
y[53] -> -2.2779517770890814, y[59] -> -178.80449406637854, <br />
y[61] -> -0.4423932640321898, y[67] -> 0.04368664901712166, <br />
y[71] -> -4.99005908979309, y[73] -> 0.2474030846661196, <br />
y[79] -> -1.1198370183785098}}, <br />
{0.5345209356631386, <br />
{y[2] -> -3535.9213959550475, y[3] -> -2.7435170695053426, <br />
y[5] -> 1841.68715251979, y[7] -> -0.08695907526585743, <br />
y[11] -> -7.560302153058244, y[13] -> -0.16070464964209902, <br />
y[17] -> -179450.3278096267, y[19] -> 0.36180370189757594, <br />
y[23] -> -11.550229002639522, y[29] -> 7.713218839254567, <br />
y[31] -> 0.0848129127512586, y[37] -> -0.4656405608004613, <br />
y[41] -> 2329.6174679703713, y[43] -> 0.42473851156047004, <br />
y[47] -> 1.5854529541054205, y[53] -> -3.1634731553535955, <br />
y[59] -> 4.7271763398480795, y[61] -> -0.8255444900077886, <br />
y[67] -> -0.01253033821081483, y[71] -> -1984.2292411392034, <br />
y[73] -> 0.08113941358865358, y[79] -> -0.4792737748163599}}, <br />
{0.5446127381623379, <br />
{y[2] -> -5.014708963920774, <br />
y[3] -> 1.6480739954794408*10^6, y[5] -> -1.8054886612263317, <br />
y[7] -> 0.8906342291461616, y[11] -> -1.707073396117987, <br />
y[13] -> 1.5628224634363812, y[17] -> -0.6611721077818026, <br />
y[19] -> 0.7750233761965257, y[23] -> -1.2435337765004315, <br />
y[29] -> -0.357284555266686, y[31] -> 1.4655841812135644, <br />
y[37] -> 11.693953407520567, y[41] -> -0.6463755063550177, <br />
y[43] -> 0.9379772581496048, y[47] -> 0.31337652569271107, <br />
y[53] -> -0.7210168316402203, y[59] -> -0.11958679378827772, <br />
y[61] -> -2.922500109233194, y[67] -> -288350.9761325468, <br />
y[71] -> 0.18284299894759842, y[73] -> -4.0197907085561515*10^6, <br />
y[79] -> 16.793919038925115}}, {0.547862473559509, <br />
{y[2] -> -6.547748625076605, y[3] -> -1.6611578546016958, <br />
y[5] -> -2.710264595234477, y[7] -> 0.3817903246124937, <br />
y[11] -> -1.332746424350139, y[13] -> 0.4227817109933836, <br />
y[17] -> -2.588003188398571, y[19] -> 0.9518471096965345, <br />
y[23] -> -1.1168350318726679, y[29] -> -2.1138721188232923, <br />
y[31] -> 1.0594308979736695, y[37] -> 0.5187896512587502, <br />
y[41] -> -0.46352692726317635, y[43] -> -1533.3644431858372, <br />
y[47] -> 735.937824142792, y[53] -> -0.5613422033234063, <br />
y[59] -> -1.5530219396118432, y[61] -> 0.28955907891150673, <br />
y[67] -> 0.7381557852332986, y[71] -> -1.0792203107683354, <br />
y[73] -> 1.4035526050070115, y[79] -> 0.7255761078124311}}, <br />
{0.5266856950717234, <br />
{y[2] -> -3459.935309033362, <br />
y[3] -> -3.4063086643916543, y[5] -> 37.04310788750117, <br />
y[7] -> -0.09690123935885808, y[11] -> -30.33949351116257, <br />
y[13] -> -0.3362756175979684, y[17] -> 5.067290751169021, <br />
y[19] -> 0.1854933403077603, y[23] -> 4013.767924077426, <br />
y[29] -> 6.758013321311452, y[31] -> 0.10637391151245976, <br />
y[37] -> -0.5388332698820616, y[41] -> 2355.2235699260627, <br />
y[43] -> 0.21017206627032822, y[47] -> 1.2870774432111025, <br />
y[53] -> 2242.906075623682, y[59] -> 3.9153456066995886, <br />
y[61] -> -0.9227311076879264, y[67] -> -0.36684003347585664, <br />
y[71] -> 5.012355770387356, y[73] -> -0.09498040322451262, <br />
y[79] -> -0.3336284179174705, y[83] -> -1.9088963090055158}}, <br />
{0.5313473007937379, <br />
{y[2] -> -4181.739559012364, <br />
y[3] -> -3.25329618970801, y[5] -> 22.642838584437005, <br />
y[7] -> -0.1270835024420529, y[11] -> -38.21863925771673, <br />
y[13] -> -0.3256175215180796, y[17] -> 5.326432709048294, <br />
y[19] -> 0.1988128336500751, y[23] -> 2550.8096908811735, <br />
y[29] -> 6.92112290656729, y[31] -> 0.09970913620864919, <br />
y[37] -> -0.5386514420799422, y[41] -> 2236.97600387609, <br />
y[43] -> 0.25154384559363013, y[47] -> 1.2353541356343836, <br />
y[53] -> 5101.731595291622, y[59] -> 3.8062097674011044, <br />
y[61] -> -0.9552297396074314, y[67] -> -0.36477843964596973, <br />
y[71] -> 4.108307130370169, y[73] -> -0.09957318641785656, <br />
y[79] -> -0.5561566460508945, y[83] -> -0.13884144425607595}}<br />
}</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2967
Omega(N)
2010-02-02T09:30:04Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}.</math><br />
<br />
A variant of the above expression for <math>\omega(N)</math> is<br />
<br />
:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math><br />
<br />
<br />
== Exact values ==<br />
<br />
These are the currently known values of <math>\Omega(N)</math><br />
<br />
<math>\Omega(1)=1</math> at <math>y_1=0</math><br />
<br />
<math>\Omega(2)=\frac{1}{2}</math> at <math>y_1=0, y_2=-\infty</math><br />
<br />
<math>\Omega(3)=\frac{1}{2}</math> at <math>y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}}</math><br />
<br />
<math>\Omega(4)=\frac{1}{2}</math> at <math>y_1=0, y_2=\infty, y_3=\infty</math><br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,84</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
A plot of the bounds for <math> N=1,...,84</math><br />
[http://abel.math.umu.se/~klasm/Data/EDP/omega-bound-plot.pdf]<br />
<br />
<nowiki><br />
<br />
</nowiki><br />
<br />
{<br />
{1, <br />
{y[1]->0}},<br />
{1/2, <br />
{y[2] -> -Infinity}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -Infinity, y[3] -> Infinity}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, <br />
{y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, <br />
{y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*10^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, <br />
{y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, <br />
{y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, <br />
{y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, <br />
{y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, <br />
{y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*10^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*10^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*10^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*10^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*10^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*10^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*10^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*10^6, <br />
y[47] -> 1.3757810922960734*10^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*10^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*10^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*10^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*10^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, <br />
{y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*10^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*10^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*10^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*10^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*10^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*10^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, <br />
{y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*10^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, <br />
{y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*10^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*10^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*10^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*10^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*10^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*10^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*10^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*10^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, <br />
{y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*10^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, <br />
{y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*10^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*10^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*10^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*10^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*10^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, <br />
{y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*10^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*10^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*10^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, <br />
{y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*10^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*10^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*10^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*10^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*10^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}},<br />
{0.5347953853151964, <br />
{y[2] -> 826616.6264389796, <br />
y[3] -> -3.2445194137959525, y[5] -> 1.1337838988884934*^6, <br />
y[7] -> -0.053183445803101505, y[11] -> 813021.9382822418, <br />
y[13] -> -0.35844167042277564, y[17] -> 5.339307410851948, <br />
y[19] -> 0.219050505622947, y[23] -> 499949.5722852526, <br />
y[29] -> -603285.5606499284, y[31] -> 0.2483912347244814, <br />
y[37] -> -0.5099629142111536, y[41] -> 1.0913022635794224*^6, <br />
y[43] -> 0.16012687574803594, y[47] -> 1.4162187543877018, <br />
y[53] -> -708771.0434949356, y[59] -> 4.926882173473444, <br />
y[61] -> -0.8700222063716009, y[67] -> -0.29467048865194995, <br />
y[71] -> 294424.5531175645, y[73] -> 0.018929724818790454}}, <br />
{0.5475609984129808, <br />
{y[2] -> -25.165389880489784, <br />
y[3] -> -2.3579309267835553, y[5] -> -8.785814809607855, <br />
y[7] -> 0.04671421855986395, y[11] -> -3.491265944223444, <br />
y[13] -> -0.04348811944806861, y[17] -> 10656.226669356573, <br />
y[19] -> 0.4981709858168526, y[23] -> -2.958123944570159, <br />
y[29] -> -12.528475056708524, y[31] -> 0.3726267566768777, <br />
y[37] -> -0.34033918120195966, y[41] -> 0.840340004423624, <br />
y[43] -> -3.3868866214535616, y[47] -> 1.675744684738037, <br />
y[53] -> -2.2779517770890814, y[59] -> -178.80449406637854, <br />
y[61] -> -0.4423932640321898, y[67] -> 0.04368664901712166, <br />
y[71] -> -4.99005908979309, y[73] -> 0.2474030846661196, <br />
y[79] -> -1.1198370183785098}}, <br />
{0.5345209356631386, <br />
{y[2] -> -3535.9213959550475, y[3] -> -2.7435170695053426, <br />
y[5] -> 1841.68715251979, y[7] -> -0.08695907526585743, <br />
y[11] -> -7.560302153058244, y[13] -> -0.16070464964209902, <br />
y[17] -> -179450.3278096267, y[19] -> 0.36180370189757594, <br />
y[23] -> -11.550229002639522, y[29] -> 7.713218839254567, <br />
y[31] -> 0.0848129127512586, y[37] -> -0.4656405608004613, <br />
y[41] -> 2329.6174679703713, y[43] -> 0.42473851156047004, <br />
y[47] -> 1.5854529541054205, y[53] -> -3.1634731553535955, <br />
y[59] -> 4.7271763398480795, y[61] -> -0.8255444900077886, <br />
y[67] -> -0.01253033821081483, y[71] -> -1984.2292411392034, <br />
y[73] -> 0.08113941358865358, y[79] -> -0.4792737748163599}}, <br />
{0.5446127381623379, <br />
{y[2] -> -5.014708963920774, <br />
y[3] -> 1.6480739954794408*^6, y[5] -> -1.8054886612263317, <br />
y[7] -> 0.8906342291461616, y[11] -> -1.707073396117987, <br />
y[13] -> 1.5628224634363812, y[17] -> -0.6611721077818026, <br />
y[19] -> 0.7750233761965257, y[23] -> -1.2435337765004315, <br />
y[29] -> -0.357284555266686, y[31] -> 1.4655841812135644, <br />
y[37] -> 11.693953407520567, y[41] -> -0.6463755063550177, <br />
y[43] -> 0.9379772581496048, y[47] -> 0.31337652569271107, <br />
y[53] -> -0.7210168316402203, y[59] -> -0.11958679378827772, <br />
y[61] -> -2.922500109233194, y[67] -> -288350.9761325468, <br />
y[71] -> 0.18284299894759842, y[73] -> -4.0197907085561515*^6, <br />
y[79] -> 16.793919038925115}}, {0.547862473559509, <br />
{y[2] -> -6.547748625076605, y[3] -> -1.6611578546016958, <br />
y[5] -> -2.710264595234477, y[7] -> 0.3817903246124937, <br />
y[11] -> -1.332746424350139, y[13] -> 0.4227817109933836, <br />
y[17] -> -2.588003188398571, y[19] -> 0.9518471096965345, <br />
y[23] -> -1.1168350318726679, y[29] -> -2.1138721188232923, <br />
y[31] -> 1.0594308979736695, y[37] -> 0.5187896512587502, <br />
y[41] -> -0.46352692726317635, y[43] -> -1533.3644431858372, <br />
y[47] -> 735.937824142792, y[53] -> -0.5613422033234063, <br />
y[59] -> -1.5530219396118432, y[61] -> 0.28955907891150673, <br />
y[67] -> 0.7381557852332986, y[71] -> -1.0792203107683354, <br />
y[73] -> 1.4035526050070115, y[79] -> 0.7255761078124311}}, <br />
{0.5266856950717234, <br />
{y[2] -> -3459.935309033362, <br />
y[3] -> -3.4063086643916543, y[5] -> 37.04310788750117, <br />
y[7] -> -0.09690123935885808, y[11] -> -30.33949351116257, <br />
y[13] -> -0.3362756175979684, y[17] -> 5.067290751169021, <br />
y[19] -> 0.1854933403077603, y[23] -> 4013.767924077426, <br />
y[29] -> 6.758013321311452, y[31] -> 0.10637391151245976, <br />
y[37] -> -0.5388332698820616, y[41] -> 2355.2235699260627, <br />
y[43] -> 0.21017206627032822, y[47] -> 1.2870774432111025, <br />
y[53] -> 2242.906075623682, y[59] -> 3.9153456066995886, <br />
y[61] -> -0.9227311076879264, y[67] -> -0.36684003347585664, <br />
y[71] -> 5.012355770387356, y[73] -> -0.09498040322451262, <br />
y[79] -> -0.3336284179174705, y[83] -> -1.9088963090055158}}, <br />
{0.5313473007937379, <br />
{y[2] -> -4181.739559012364, <br />
y[3] -> -3.25329618970801, y[5] -> 22.642838584437005, <br />
y[7] -> -0.1270835024420529, y[11] -> -38.21863925771673, <br />
y[13] -> -0.3256175215180796, y[17] -> 5.326432709048294, <br />
y[19] -> 0.1988128336500751, y[23] -> 2550.8096908811735, <br />
y[29] -> 6.92112290656729, y[31] -> 0.09970913620864919, <br />
y[37] -> -0.5386514420799422, y[41] -> 2236.97600387609, <br />
y[43] -> 0.25154384559363013, y[47] -> 1.2353541356343836, <br />
y[53] -> 5101.731595291622, y[59] -> 3.8062097674011044, <br />
y[61] -> -0.9552297396074314, y[67] -> -0.36477843964596973, <br />
y[71] -> 4.108307130370169, y[73] -> -0.09957318641785656, <br />
y[79] -> -0.5561566460508945, y[83] -> -0.13884144425607595}}<br />
}</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2966
Omega(N)
2010-02-02T09:11:22Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}.</math><br />
<br />
A variant of the above expression for <math>\omega(N)</math> is<br />
<br />
:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math><br />
<br />
<br />
== Exact values ==<br />
<br />
These are the currently known values of <math>\Omega(N)</math><br />
<br />
<math>\Omega(1)=1</math> at <math>y_1=0</math><br />
<br />
<math>\Omega(2)=\frac{1}{2}</math> at <math>y_1=0, y_2=-\infty</math><br />
<br />
<math>\Omega(3)=\frac{1}{2}</math> at <math>y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}}</math><br />
<br />
<math>\Omega(4)=\frac{1}{2}</math> <br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,84</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
A plot of the bounds for <math> N=1,...,84</math><br />
[http://abel.math.umu.se/~klasm/Data/EDP/omega-bound-plot.pdf]<br />
<br />
<nowiki><br />
<br />
</nowiki><br />
<br />
{<br />
{1, <br />
{y[1]->0}},<br />
{1/2, <br />
{y[2] -> -Infinity}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -Infinity, y[3] -> Infinity}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, <br />
{y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, <br />
{y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*10^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, <br />
{y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, <br />
{y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, <br />
{y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, <br />
{y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, <br />
{y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*10^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*10^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*10^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*10^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*10^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*10^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*10^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*10^6, <br />
y[47] -> 1.3757810922960734*10^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*10^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*10^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*10^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*10^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, <br />
{y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*10^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*10^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*10^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*10^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*10^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*10^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, <br />
{y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*10^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, <br />
{y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*10^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*10^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*10^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*10^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*10^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*10^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*10^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*10^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, <br />
{y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*10^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, <br />
{y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*10^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*10^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*10^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*10^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*10^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, <br />
{y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*10^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*10^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*10^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, <br />
{y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*10^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*10^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*10^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*10^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*10^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}},<br />
{0.5347953853151964, <br />
{y[2] -> 826616.6264389796, <br />
y[3] -> -3.2445194137959525, y[5] -> 1.1337838988884934*^6, <br />
y[7] -> -0.053183445803101505, y[11] -> 813021.9382822418, <br />
y[13] -> -0.35844167042277564, y[17] -> 5.339307410851948, <br />
y[19] -> 0.219050505622947, y[23] -> 499949.5722852526, <br />
y[29] -> -603285.5606499284, y[31] -> 0.2483912347244814, <br />
y[37] -> -0.5099629142111536, y[41] -> 1.0913022635794224*^6, <br />
y[43] -> 0.16012687574803594, y[47] -> 1.4162187543877018, <br />
y[53] -> -708771.0434949356, y[59] -> 4.926882173473444, <br />
y[61] -> -0.8700222063716009, y[67] -> -0.29467048865194995, <br />
y[71] -> 294424.5531175645, y[73] -> 0.018929724818790454}}, <br />
{0.5475609984129808, <br />
{y[2] -> -25.165389880489784, <br />
y[3] -> -2.3579309267835553, y[5] -> -8.785814809607855, <br />
y[7] -> 0.04671421855986395, y[11] -> -3.491265944223444, <br />
y[13] -> -0.04348811944806861, y[17] -> 10656.226669356573, <br />
y[19] -> 0.4981709858168526, y[23] -> -2.958123944570159, <br />
y[29] -> -12.528475056708524, y[31] -> 0.3726267566768777, <br />
y[37] -> -0.34033918120195966, y[41] -> 0.840340004423624, <br />
y[43] -> -3.3868866214535616, y[47] -> 1.675744684738037, <br />
y[53] -> -2.2779517770890814, y[59] -> -178.80449406637854, <br />
y[61] -> -0.4423932640321898, y[67] -> 0.04368664901712166, <br />
y[71] -> -4.99005908979309, y[73] -> 0.2474030846661196, <br />
y[79] -> -1.1198370183785098}}, <br />
{0.5345209356631386, <br />
{y[2] -> -3535.9213959550475, y[3] -> -2.7435170695053426, <br />
y[5] -> 1841.68715251979, y[7] -> -0.08695907526585743, <br />
y[11] -> -7.560302153058244, y[13] -> -0.16070464964209902, <br />
y[17] -> -179450.3278096267, y[19] -> 0.36180370189757594, <br />
y[23] -> -11.550229002639522, y[29] -> 7.713218839254567, <br />
y[31] -> 0.0848129127512586, y[37] -> -0.4656405608004613, <br />
y[41] -> 2329.6174679703713, y[43] -> 0.42473851156047004, <br />
y[47] -> 1.5854529541054205, y[53] -> -3.1634731553535955, <br />
y[59] -> 4.7271763398480795, y[61] -> -0.8255444900077886, <br />
y[67] -> -0.01253033821081483, y[71] -> -1984.2292411392034, <br />
y[73] -> 0.08113941358865358, y[79] -> -0.4792737748163599}}, <br />
{0.5446127381623379, <br />
{y[2] -> -5.014708963920774, <br />
y[3] -> 1.6480739954794408*^6, y[5] -> -1.8054886612263317, <br />
y[7] -> 0.8906342291461616, y[11] -> -1.707073396117987, <br />
y[13] -> 1.5628224634363812, y[17] -> -0.6611721077818026, <br />
y[19] -> 0.7750233761965257, y[23] -> -1.2435337765004315, <br />
y[29] -> -0.357284555266686, y[31] -> 1.4655841812135644, <br />
y[37] -> 11.693953407520567, y[41] -> -0.6463755063550177, <br />
y[43] -> 0.9379772581496048, y[47] -> 0.31337652569271107, <br />
y[53] -> -0.7210168316402203, y[59] -> -0.11958679378827772, <br />
y[61] -> -2.922500109233194, y[67] -> -288350.9761325468, <br />
y[71] -> 0.18284299894759842, y[73] -> -4.0197907085561515*^6, <br />
y[79] -> 16.793919038925115}}, {0.547862473559509, <br />
{y[2] -> -6.547748625076605, y[3] -> -1.6611578546016958, <br />
y[5] -> -2.710264595234477, y[7] -> 0.3817903246124937, <br />
y[11] -> -1.332746424350139, y[13] -> 0.4227817109933836, <br />
y[17] -> -2.588003188398571, y[19] -> 0.9518471096965345, <br />
y[23] -> -1.1168350318726679, y[29] -> -2.1138721188232923, <br />
y[31] -> 1.0594308979736695, y[37] -> 0.5187896512587502, <br />
y[41] -> -0.46352692726317635, y[43] -> -1533.3644431858372, <br />
y[47] -> 735.937824142792, y[53] -> -0.5613422033234063, <br />
y[59] -> -1.5530219396118432, y[61] -> 0.28955907891150673, <br />
y[67] -> 0.7381557852332986, y[71] -> -1.0792203107683354, <br />
y[73] -> 1.4035526050070115, y[79] -> 0.7255761078124311}}, <br />
{0.5266856950717234, <br />
{y[2] -> -3459.935309033362, <br />
y[3] -> -3.4063086643916543, y[5] -> 37.04310788750117, <br />
y[7] -> -0.09690123935885808, y[11] -> -30.33949351116257, <br />
y[13] -> -0.3362756175979684, y[17] -> 5.067290751169021, <br />
y[19] -> 0.1854933403077603, y[23] -> 4013.767924077426, <br />
y[29] -> 6.758013321311452, y[31] -> 0.10637391151245976, <br />
y[37] -> -0.5388332698820616, y[41] -> 2355.2235699260627, <br />
y[43] -> 0.21017206627032822, y[47] -> 1.2870774432111025, <br />
y[53] -> 2242.906075623682, y[59] -> 3.9153456066995886, <br />
y[61] -> -0.9227311076879264, y[67] -> -0.36684003347585664, <br />
y[71] -> 5.012355770387356, y[73] -> -0.09498040322451262, <br />
y[79] -> -0.3336284179174705, y[83] -> -1.9088963090055158}}, <br />
{0.5313473007937379, <br />
{y[2] -> -4181.739559012364, <br />
y[3] -> -3.25329618970801, y[5] -> 22.642838584437005, <br />
y[7] -> -0.1270835024420529, y[11] -> -38.21863925771673, <br />
y[13] -> -0.3256175215180796, y[17] -> 5.326432709048294, <br />
y[19] -> 0.1988128336500751, y[23] -> 2550.8096908811735, <br />
y[29] -> 6.92112290656729, y[31] -> 0.09970913620864919, <br />
y[37] -> -0.5386514420799422, y[41] -> 2236.97600387609, <br />
y[43] -> 0.25154384559363013, y[47] -> 1.2353541356343836, <br />
y[53] -> 5101.731595291622, y[59] -> 3.8062097674011044, <br />
y[61] -> -0.9552297396074314, y[67] -> -0.36477843964596973, <br />
y[71] -> 4.108307130370169, y[73] -> -0.09957318641785656, <br />
y[79] -> -0.5561566460508945, y[83] -> -0.13884144425607595}}<br />
}</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2965
Omega(N)
2010-02-02T08:55:07Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}.</math><br />
<br />
A variant of the above expression for <math>\omega(N)</math> is<br />
<br />
:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math><br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,84</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
A plot of the bounds for <math> N=1,...,84</math><br />
[http://abel.math.umu.se/~klasm/Data/EDP/omega-bound-plot.pdf]<br />
<br />
<nowiki><br />
<br />
</nowiki><br />
<br />
{<br />
{1, <br />
{y[1]->0}},<br />
{1/2, <br />
{y[2] -> -Infinity}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -Infinity, y[3] -> Infinity}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, <br />
{y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, <br />
{y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*10^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, <br />
{y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, <br />
{y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, <br />
{y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, <br />
{y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, <br />
{y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*10^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*10^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*10^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*10^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*10^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*10^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*10^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*10^6, <br />
y[47] -> 1.3757810922960734*10^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*10^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*10^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*10^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*10^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, <br />
{y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*10^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*10^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*10^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*10^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*10^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*10^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, <br />
{y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*10^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, <br />
{y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*10^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*10^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*10^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*10^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*10^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*10^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*10^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*10^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, <br />
{y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*10^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, <br />
{y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*10^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*10^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*10^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*10^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*10^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, <br />
{y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*10^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*10^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*10^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, <br />
{y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*10^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*10^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*10^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*10^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*10^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}},<br />
{0.5347953853151964, <br />
{y[2] -> 826616.6264389796, <br />
y[3] -> -3.2445194137959525, y[5] -> 1.1337838988884934*^6, <br />
y[7] -> -0.053183445803101505, y[11] -> 813021.9382822418, <br />
y[13] -> -0.35844167042277564, y[17] -> 5.339307410851948, <br />
y[19] -> 0.219050505622947, y[23] -> 499949.5722852526, <br />
y[29] -> -603285.5606499284, y[31] -> 0.2483912347244814, <br />
y[37] -> -0.5099629142111536, y[41] -> 1.0913022635794224*^6, <br />
y[43] -> 0.16012687574803594, y[47] -> 1.4162187543877018, <br />
y[53] -> -708771.0434949356, y[59] -> 4.926882173473444, <br />
y[61] -> -0.8700222063716009, y[67] -> -0.29467048865194995, <br />
y[71] -> 294424.5531175645, y[73] -> 0.018929724818790454}}, <br />
{0.5475609984129808, <br />
{y[2] -> -25.165389880489784, <br />
y[3] -> -2.3579309267835553, y[5] -> -8.785814809607855, <br />
y[7] -> 0.04671421855986395, y[11] -> -3.491265944223444, <br />
y[13] -> -0.04348811944806861, y[17] -> 10656.226669356573, <br />
y[19] -> 0.4981709858168526, y[23] -> -2.958123944570159, <br />
y[29] -> -12.528475056708524, y[31] -> 0.3726267566768777, <br />
y[37] -> -0.34033918120195966, y[41] -> 0.840340004423624, <br />
y[43] -> -3.3868866214535616, y[47] -> 1.675744684738037, <br />
y[53] -> -2.2779517770890814, y[59] -> -178.80449406637854, <br />
y[61] -> -0.4423932640321898, y[67] -> 0.04368664901712166, <br />
y[71] -> -4.99005908979309, y[73] -> 0.2474030846661196, <br />
y[79] -> -1.1198370183785098}}, <br />
{0.5345209356631386, <br />
{y[2] -> -3535.9213959550475, y[3] -> -2.7435170695053426, <br />
y[5] -> 1841.68715251979, y[7] -> -0.08695907526585743, <br />
y[11] -> -7.560302153058244, y[13] -> -0.16070464964209902, <br />
y[17] -> -179450.3278096267, y[19] -> 0.36180370189757594, <br />
y[23] -> -11.550229002639522, y[29] -> 7.713218839254567, <br />
y[31] -> 0.0848129127512586, y[37] -> -0.4656405608004613, <br />
y[41] -> 2329.6174679703713, y[43] -> 0.42473851156047004, <br />
y[47] -> 1.5854529541054205, y[53] -> -3.1634731553535955, <br />
y[59] -> 4.7271763398480795, y[61] -> -0.8255444900077886, <br />
y[67] -> -0.01253033821081483, y[71] -> -1984.2292411392034, <br />
y[73] -> 0.08113941358865358, y[79] -> -0.4792737748163599}}, <br />
{0.5446127381623379, <br />
{y[2] -> -5.014708963920774, <br />
y[3] -> 1.6480739954794408*^6, y[5] -> -1.8054886612263317, <br />
y[7] -> 0.8906342291461616, y[11] -> -1.707073396117987, <br />
y[13] -> 1.5628224634363812, y[17] -> -0.6611721077818026, <br />
y[19] -> 0.7750233761965257, y[23] -> -1.2435337765004315, <br />
y[29] -> -0.357284555266686, y[31] -> 1.4655841812135644, <br />
y[37] -> 11.693953407520567, y[41] -> -0.6463755063550177, <br />
y[43] -> 0.9379772581496048, y[47] -> 0.31337652569271107, <br />
y[53] -> -0.7210168316402203, y[59] -> -0.11958679378827772, <br />
y[61] -> -2.922500109233194, y[67] -> -288350.9761325468, <br />
y[71] -> 0.18284299894759842, y[73] -> -4.0197907085561515*^6, <br />
y[79] -> 16.793919038925115}}, {0.547862473559509, <br />
{y[2] -> -6.547748625076605, y[3] -> -1.6611578546016958, <br />
y[5] -> -2.710264595234477, y[7] -> 0.3817903246124937, <br />
y[11] -> -1.332746424350139, y[13] -> 0.4227817109933836, <br />
y[17] -> -2.588003188398571, y[19] -> 0.9518471096965345, <br />
y[23] -> -1.1168350318726679, y[29] -> -2.1138721188232923, <br />
y[31] -> 1.0594308979736695, y[37] -> 0.5187896512587502, <br />
y[41] -> -0.46352692726317635, y[43] -> -1533.3644431858372, <br />
y[47] -> 735.937824142792, y[53] -> -0.5613422033234063, <br />
y[59] -> -1.5530219396118432, y[61] -> 0.28955907891150673, <br />
y[67] -> 0.7381557852332986, y[71] -> -1.0792203107683354, <br />
y[73] -> 1.4035526050070115, y[79] -> 0.7255761078124311}}, <br />
{0.5266856950717234, <br />
{y[2] -> -3459.935309033362, <br />
y[3] -> -3.4063086643916543, y[5] -> 37.04310788750117, <br />
y[7] -> -0.09690123935885808, y[11] -> -30.33949351116257, <br />
y[13] -> -0.3362756175979684, y[17] -> 5.067290751169021, <br />
y[19] -> 0.1854933403077603, y[23] -> 4013.767924077426, <br />
y[29] -> 6.758013321311452, y[31] -> 0.10637391151245976, <br />
y[37] -> -0.5388332698820616, y[41] -> 2355.2235699260627, <br />
y[43] -> 0.21017206627032822, y[47] -> 1.2870774432111025, <br />
y[53] -> 2242.906075623682, y[59] -> 3.9153456066995886, <br />
y[61] -> -0.9227311076879264, y[67] -> -0.36684003347585664, <br />
y[71] -> 5.012355770387356, y[73] -> -0.09498040322451262, <br />
y[79] -> -0.3336284179174705, y[83] -> -1.9088963090055158}}, <br />
{0.5313473007937379, <br />
{y[2] -> -4181.739559012364, <br />
y[3] -> -3.25329618970801, y[5] -> 22.642838584437005, <br />
y[7] -> -0.1270835024420529, y[11] -> -38.21863925771673, <br />
y[13] -> -0.3256175215180796, y[17] -> 5.326432709048294, <br />
y[19] -> 0.1988128336500751, y[23] -> 2550.8096908811735, <br />
y[29] -> 6.92112290656729, y[31] -> 0.09970913620864919, <br />
y[37] -> -0.5386514420799422, y[41] -> 2236.97600387609, <br />
y[43] -> 0.25154384559363013, y[47] -> 1.2353541356343836, <br />
y[53] -> 5101.731595291622, y[59] -> 3.8062097674011044, <br />
y[61] -> -0.9552297396074314, y[67] -> -0.36477843964596973, <br />
y[71] -> 4.108307130370169, y[73] -> -0.09957318641785656, <br />
y[79] -> -0.5561566460508945, y[83] -> -0.13884144425607595}}<br />
}</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2964
Omega(N)
2010-02-02T08:19:11Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}.</math><br />
<br />
A variant of the above expression for <math>\omega(N)</math> is<br />
<br />
:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math><br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
<nowiki><br />
<br />
</nowiki><br />
<br />
{<br />
{1, <br />
{y[1]->0}},<br />
{1/2, <br />
{y[2] -> -Infinity}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -Infinity, y[3] -> Infinity}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, <br />
{y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, <br />
{y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*10^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, <br />
{y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, <br />
{y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, <br />
{y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, <br />
{y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, <br />
{y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*10^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*10^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*10^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*10^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*10^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*10^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*10^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*10^6, <br />
y[47] -> 1.3757810922960734*10^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*10^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*10^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*10^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*10^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, <br />
{y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*10^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*10^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*10^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*10^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*10^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*10^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, <br />
{y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*10^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, <br />
{y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*10^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*10^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*10^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*10^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*10^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*10^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*10^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*10^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, <br />
{y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*10^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, <br />
{y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*10^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*10^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*10^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*10^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*10^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, <br />
{y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*10^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*10^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*10^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, <br />
{y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*10^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*10^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*10^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*10^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*10^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}},<br />
{0.5347953853151964, <br />
{y[2] -> 826616.6264389796, <br />
y[3] -> -3.2445194137959525, y[5] -> 1.1337838988884934*^6, <br />
y[7] -> -0.053183445803101505, y[11] -> 813021.9382822418, <br />
y[13] -> -0.35844167042277564, y[17] -> 5.339307410851948, <br />
y[19] -> 0.219050505622947, y[23] -> 499949.5722852526, <br />
y[29] -> -603285.5606499284, y[31] -> 0.2483912347244814, <br />
y[37] -> -0.5099629142111536, y[41] -> 1.0913022635794224*^6, <br />
y[43] -> 0.16012687574803594, y[47] -> 1.4162187543877018, <br />
y[53] -> -708771.0434949356, y[59] -> 4.926882173473444, <br />
y[61] -> -0.8700222063716009, y[67] -> -0.29467048865194995, <br />
y[71] -> 294424.5531175645, y[73] -> 0.018929724818790454}}, <br />
{0.5475609984129808, <br />
{y[2] -> -25.165389880489784, <br />
y[3] -> -2.3579309267835553, y[5] -> -8.785814809607855, <br />
y[7] -> 0.04671421855986395, y[11] -> -3.491265944223444, <br />
y[13] -> -0.04348811944806861, y[17] -> 10656.226669356573, <br />
y[19] -> 0.4981709858168526, y[23] -> -2.958123944570159, <br />
y[29] -> -12.528475056708524, y[31] -> 0.3726267566768777, <br />
y[37] -> -0.34033918120195966, y[41] -> 0.840340004423624, <br />
y[43] -> -3.3868866214535616, y[47] -> 1.675744684738037, <br />
y[53] -> -2.2779517770890814, y[59] -> -178.80449406637854, <br />
y[61] -> -0.4423932640321898, y[67] -> 0.04368664901712166, <br />
y[71] -> -4.99005908979309, y[73] -> 0.2474030846661196, <br />
y[79] -> -1.1198370183785098}}, <br />
{0.5345209356631386, <br />
{y[2] -> -3535.9213959550475, y[3] -> -2.7435170695053426, <br />
y[5] -> 1841.68715251979, y[7] -> -0.08695907526585743, <br />
y[11] -> -7.560302153058244, y[13] -> -0.16070464964209902, <br />
y[17] -> -179450.3278096267, y[19] -> 0.36180370189757594, <br />
y[23] -> -11.550229002639522, y[29] -> 7.713218839254567, <br />
y[31] -> 0.0848129127512586, y[37] -> -0.4656405608004613, <br />
y[41] -> 2329.6174679703713, y[43] -> 0.42473851156047004, <br />
y[47] -> 1.5854529541054205, y[53] -> -3.1634731553535955, <br />
y[59] -> 4.7271763398480795, y[61] -> -0.8255444900077886, <br />
y[67] -> -0.01253033821081483, y[71] -> -1984.2292411392034, <br />
y[73] -> 0.08113941358865358, y[79] -> -0.4792737748163599}}, <br />
{0.5446127381623379, <br />
{y[2] -> -5.014708963920774, <br />
y[3] -> 1.6480739954794408*^6, y[5] -> -1.8054886612263317, <br />
y[7] -> 0.8906342291461616, y[11] -> -1.707073396117987, <br />
y[13] -> 1.5628224634363812, y[17] -> -0.6611721077818026, <br />
y[19] -> 0.7750233761965257, y[23] -> -1.2435337765004315, <br />
y[29] -> -0.357284555266686, y[31] -> 1.4655841812135644, <br />
y[37] -> 11.693953407520567, y[41] -> -0.6463755063550177, <br />
y[43] -> 0.9379772581496048, y[47] -> 0.31337652569271107, <br />
y[53] -> -0.7210168316402203, y[59] -> -0.11958679378827772, <br />
y[61] -> -2.922500109233194, y[67] -> -288350.9761325468, <br />
y[71] -> 0.18284299894759842, y[73] -> -4.0197907085561515*^6, <br />
y[79] -> 16.793919038925115}}, {0.547862473559509, <br />
{y[2] -> -6.547748625076605, y[3] -> -1.6611578546016958, <br />
y[5] -> -2.710264595234477, y[7] -> 0.3817903246124937, <br />
y[11] -> -1.332746424350139, y[13] -> 0.4227817109933836, <br />
y[17] -> -2.588003188398571, y[19] -> 0.9518471096965345, <br />
y[23] -> -1.1168350318726679, y[29] -> -2.1138721188232923, <br />
y[31] -> 1.0594308979736695, y[37] -> 0.5187896512587502, <br />
y[41] -> -0.46352692726317635, y[43] -> -1533.3644431858372, <br />
y[47] -> 735.937824142792, y[53] -> -0.5613422033234063, <br />
y[59] -> -1.5530219396118432, y[61] -> 0.28955907891150673, <br />
y[67] -> 0.7381557852332986, y[71] -> -1.0792203107683354, <br />
y[73] -> 1.4035526050070115, y[79] -> 0.7255761078124311}}, <br />
{0.5266856950717234, <br />
{y[2] -> -3459.935309033362, <br />
y[3] -> -3.4063086643916543, y[5] -> 37.04310788750117, <br />
y[7] -> -0.09690123935885808, y[11] -> -30.33949351116257, <br />
y[13] -> -0.3362756175979684, y[17] -> 5.067290751169021, <br />
y[19] -> 0.1854933403077603, y[23] -> 4013.767924077426, <br />
y[29] -> 6.758013321311452, y[31] -> 0.10637391151245976, <br />
y[37] -> -0.5388332698820616, y[41] -> 2355.2235699260627, <br />
y[43] -> 0.21017206627032822, y[47] -> 1.2870774432111025, <br />
y[53] -> 2242.906075623682, y[59] -> 3.9153456066995886, <br />
y[61] -> -0.9227311076879264, y[67] -> -0.36684003347585664, <br />
y[71] -> 5.012355770387356, y[73] -> -0.09498040322451262, <br />
y[79] -> -0.3336284179174705, y[83] -> -1.9088963090055158}}, <br />
{0.5313473007937379, <br />
{y[2] -> -4181.739559012364, <br />
y[3] -> -3.25329618970801, y[5] -> 22.642838584437005, <br />
y[7] -> -0.1270835024420529, y[11] -> -38.21863925771673, <br />
y[13] -> -0.3256175215180796, y[17] -> 5.326432709048294, <br />
y[19] -> 0.1988128336500751, y[23] -> 2550.8096908811735, <br />
y[29] -> 6.92112290656729, y[31] -> 0.09970913620864919, <br />
y[37] -> -0.5386514420799422, y[41] -> 2236.97600387609, <br />
y[43] -> 0.25154384559363013, y[47] -> 1.2353541356343836, <br />
y[53] -> 5101.731595291622, y[59] -> 3.8062097674011044, <br />
y[61] -> -0.9552297396074314, y[67] -> -0.36477843964596973, <br />
y[71] -> 4.108307130370169, y[73] -> -0.09957318641785656, <br />
y[79] -> -0.5561566460508945, y[83] -> -0.13884144425607595}}<br />
}</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2925
Omega(N)
2010-02-01T19:22:03Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math></math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-j) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N \frac{(n-j+1) \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}</math><br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
<nowiki><br />
<br />
</nowiki><br />
<br />
{<br />
{1, <br />
{y[1]->0}},<br />
{0.5, <br />
{y[2] -> -1.462885206912493*10^8}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -3.910648170382393*10^8, y[3] -> 7.310011357069776*10^8}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, <br />
{y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, <br />
{y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*10^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, <br />
{y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, <br />
{y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, <br />
{y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, <br />
{y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, <br />
{y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*10^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*10^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*10^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*10^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*10^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*10^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*10^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*10^6, <br />
y[47] -> 1.3757810922960734*10^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*10^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*10^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*10^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*10^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, <br />
{y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*10^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*10^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*10^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*10^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*10^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*10^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, <br />
{y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*10^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, <br />
{y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*10^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*10^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*10^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*10^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*10^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*10^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*10^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*10^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, <br />
{y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*10^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, <br />
{y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*10^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*10^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*10^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*10^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*10^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, <br />
{y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*10^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*10^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*10^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, <br />
{y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*10^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*10^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*10^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*10^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*10^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}}<br />
}</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2924
Omega(N)
2010-02-01T19:20:42Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math></math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-j) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N \frac{(n-j+1) \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}</math><br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
<nowiki><br />
<br />
</nowiki><br />
<br />
{<br />
<br />
{0.5, <br />
{y[2] -> -1.462885206912493*10^8}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -3.910648170382393*10^8, y[3] -> 7.310011357069776*10^8}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, <br />
{y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, <br />
{y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*10^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, <br />
{y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, <br />
{y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, <br />
{y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, <br />
{y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, <br />
{y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*10^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*10^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*10^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*10^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*10^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*10^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*10^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*10^6, <br />
y[47] -> 1.3757810922960734*10^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*10^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*10^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*10^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*10^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, <br />
{y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*10^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*10^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*10^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*10^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*10^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*10^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, <br />
{y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*10^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, <br />
{y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*10^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*10^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*10^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*10^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*10^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*10^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*10^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*10^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, <br />
{y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*10^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, <br />
{y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*10^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*10^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*10^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*10^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*10^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, <br />
{y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*10^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*10^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*10^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, <br />
{y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*10^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*10^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*10^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*10^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*10^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}}<br />
}</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2923
Omega(N)
2010-02-01T19:10:45Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math></math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-j) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N \frac{(n-j+1) \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}</math><br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
<nowiki><br />
{<br />
{0.5, {y[2] -> -1.462885206912493*^8}}, <br />
{0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, <br />
{y[2] -> -3.910648170382393*^8, y[3] -> 7.310011357069776*^8}}, <br />
{0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}}, <br />
{0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}}, <br />
{0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, <br />
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, <br />
{0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, <br />
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, <br />
{0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, <br />
{0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, <br />
{0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, <br />
{y[2] -> -8.444874723393042*^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, <br />
{0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, <br />
{0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*^6, <br />
y[17] -> -1.3074794690981901}}, <br />
{0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, <br />
{0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, {y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, <br />
{0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, {y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, <br />
{0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, <br />
{0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*^6}}, <br />
{0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, <br />
{0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, <br />
{0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, <br />
{0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, <br />
{0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, {y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, <br />
{0.5098368447944086, <br />
{y[2] -> -4.295234386315433*^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, <br />
{0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, <br />
{0.49555821568780584, <br />
{y[2] -> -5.066246086172957*^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, <br />
{0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*^7, <br />
y[31] -> -0.8641525290683024}}, <br />
{0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, <br />
{0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, <br />
{0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, {y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, <br />
{0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, {y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, <br />
{0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, <br />
{0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, <br />
{0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, {y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, <br />
{0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, {y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, <br />
{0.5538527358037998, <br />
{y[2] -> -6.019028440570796*^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, <br />
{0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, <br />
{0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*^6, <br />
y[47] -> -1.3892438499060007}}, <br />
{0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*^6, <br />
y[47] -> 1.3757810922960734*^7}}, <br />
{0.544623917879015, <br />
{y[2] -> -1.6079312057861998*^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, <br />
{0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, <br />
{0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, {y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, <br />
{0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, {y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, <br />
{0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, {y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, <br />
{0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, <br />
{0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, <br />
{0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, {y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, <br />
{0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, {y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*^6, <br />
y[61] -> -0.6400367372252429}}, <br />
{0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, {y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, <br />
{0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, <br />
{0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, <br />
{0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, <br />
{0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, <br />
{0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
{0.5573973761655405, {y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, <br />
{0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, <br />
{0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, <br />
{0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, <br />
{0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, <br />
{0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}}<br />
}<br />
</nowiki></div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2922
Omega(N)
2010-02-01T19:00:01Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math></math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-j) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N \frac{(n-j+1) \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}</math><br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
<nowiki><br />
{<br />
{0.5, {y[2] -> -1.462885206912493*^8}}, {0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, {y[2] -> -3.910648170382393*^8, <br />
y[3] -> 7.310011357069776*^8}}, {0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, <br />
y[5] -> 1.0297311434019392}}, {0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, <br />
y[5] -> -2.482521195509378}}, {0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, {y[2] -> -6.302328081648096, <br />
y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, {0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, {y[2] -> -11.150737464977489, <br />
y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, {0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, {0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, {0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, {y[2] -> -8.444874723393042*^6, <br />
y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, {0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, {0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*^6, <br />
y[17] -> -1.3074794690981901}}, {0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, {0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, {y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, {0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, {y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, {0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, {0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*^6}}, {0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, {0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, {0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, {0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, {0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, {y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, {0.5098368447944086, <br />
{y[2] -> -4.295234386315433*^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, {0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, {0.49555821568780584, <br />
{y[2] -> -5.066246086172957*^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, {0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*^7, <br />
y[31] -> -0.8641525290683024}}, {0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, {0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, {0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, {y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, {0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, {y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, {0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, {0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, {0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, {y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, {0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, {y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, {0.5538527358037998, <br />
{y[2] -> -6.019028440570796*^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, {0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, {0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*^6, <br />
y[47] -> -1.3892438499060007}}, {0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*^6, <br />
y[47] -> 1.3757810922960734*^7}}, {0.544623917879015, <br />
{y[2] -> -1.6079312057861998*^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, {0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, {0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, {y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, {0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, {y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, {0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, {y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, {0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, {0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, {0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, {y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, {0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, {y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*^6, <br />
y[61] -> -0.6400367372252429}}, {0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, {y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, {0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, {0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, {0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, {0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, {0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}},<br />
,{0.5573973761655405, {y[2] -> -2.3219438426126686, <br />
y[3] -> 1.57314510283561, y[5] -> -0.7969756633631068, <br />
y[7] -> 2.475949629405956, y[11] -> 0.11714291837133678, <br />
y[13] -> -4.592403466597099*^6, y[17] -> -0.09157610270015053, <br />
y[19] -> -3.0271178867960975, y[23] -> 0.6781963740792064, <br />
y[29] -> 1.8116961232139452, y[31] -> -0.720498237137021, <br />
y[37] -> 3.584361117953021*^6, y[41] -> 0.09818217055201135, <br />
y[43] -> -0.670743815371697, y[47] -> 1.3906057757176344, <br />
y[53] -> 1.2011297119396716, y[59] -> -7.008629047553877*^6, <br />
y[61] -> -0.6221520064455753, y[67] -> -3.2587014793232143, <br />
y[71] -> 0.3398670199613038}}, {0.5339612466171857, <br />
{y[2] -> 1.0228307911152838*^6, y[3] -> -3.182428542884625, <br />
y[5] -> 11.890077722642841, y[7] -> -0.11140413880261893, <br />
y[11] -> 692845.4573913107, y[13] -> -0.38484388508692646, <br />
y[17] -> 4.341209952301811, y[19] -> 0.2656258975978876, <br />
y[23] -> 3.4346255374415917*^6, y[29] -> -390247.5015888775, <br />
y[31] -> 0.2038068617269559, y[37] -> -0.451817441936389, <br />
y[41] -> 46897.88938120262, y[43] -> 0.21583707859265933, <br />
y[47] -> 1.3198397220473823, y[53] -> -210028.69669027862, <br />
y[59] -> 3.6505639032429418, y[61] -> -1.172691080849604, <br />
y[67] -> -0.4347480730614195, y[71] -> 7.020023800941956, <br />
y[73] -> 0.5487641897289582}}, {0.5353763772210614, <br />
{y[2] -> 470309.5183097746, y[3] -> -3.289242855457388, <br />
y[5] -> 14.138715606028944, y[7] -> -0.0857645248971096, <br />
y[11] -> 245376.27364463784, y[13] -> -0.3902174781223069, <br />
y[17] -> 4.189603762669572, y[19] -> 0.2575994244234302, <br />
y[23] -> 162163.1602140669, y[29] -> -190792.43385599708, <br />
y[31] -> 0.21327761088925834, y[37] -> -0.4337781751476201, <br />
y[41] -> 259028.86638774566, y[43] -> 0.17798223901379964, <br />
y[47] -> 1.4021884315223392, y[53] -> -113413.47319852018, <br />
y[59] -> 3.489295385976381, y[61] -> -1.205411995765, <br />
y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, <br />
y[73] -> 0.2220413097490152}}, {0.5386875035297409, <br />
{y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, <br />
y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, <br />
y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, <br />
y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, <br />
y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, <br />
y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, <br />
y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, <br />
y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, <br />
y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, <br />
y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, <br />
y[73] -> 0.0007043557151270179}}, {0.5579498988797087, <br />
{y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, <br />
y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, <br />
y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, <br />
y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, <br />
y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, <br />
y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, <br />
y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, <br />
y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, <br />
y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, <br />
y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, <br />
y[73] -> 0.5613454008457452}}, {0.5391983432183001, <br />
{y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, <br />
y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, <br />
y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, <br />
y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, <br />
y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, <br />
y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, <br />
y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, <br />
y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, <br />
y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, <br />
y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, <br />
y[73] -> -0.08134651616351249}}<br />
}<br />
</nowiki></div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2917
Omega(N)
2010-02-01T12:42:21Z
<p>Klasm: </p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math></math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-j) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N \frac{(n-j+1) \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}</math><br />
<br />
<br />
== Upper bounds for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
<nowiki><br />
{<br />
{0.5, {y[2] -> -1.462885206912493*^8}}, {0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, {y[2] -> -3.910648170382393*^8, <br />
y[3] -> 7.310011357069776*^8}}, {0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, <br />
y[5] -> 1.0297311434019392}}, {0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, <br />
y[5] -> -2.482521195509378}}, {0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, {y[2] -> -6.302328081648096, <br />
y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, {0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, {y[2] -> -11.150737464977489, <br />
y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, {0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, {0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, {0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, {y[2] -> -8.444874723393042*^6, <br />
y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, {0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, {0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*^6, <br />
y[17] -> -1.3074794690981901}}, {0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, {0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, {y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, {0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, {y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, {0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, {0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*^6}}, {0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, {0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, {0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, {0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, {0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, {y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, {0.5098368447944086, <br />
{y[2] -> -4.295234386315433*^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, {0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, {0.49555821568780584, <br />
{y[2] -> -5.066246086172957*^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, {0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*^7, <br />
y[31] -> -0.8641525290683024}}, {0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, {0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, {0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, {y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, {0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, {y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, {0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, {0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, {0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, {y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, {0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, {y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, {0.5538527358037998, <br />
{y[2] -> -6.019028440570796*^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, {0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, {0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*^6, <br />
y[47] -> -1.3892438499060007}}, {0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*^6, <br />
y[47] -> 1.3757810922960734*^7}}, {0.544623917879015, <br />
{y[2] -> -1.6079312057861998*^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, {0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, {0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, {y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, {0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, {y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, {0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, {y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, {0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, {0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, {0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, {y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, {0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, {y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*^6, <br />
y[61] -> -0.6400367372252429}}, {0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, {y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, {0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, {0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, {0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, {0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, {0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}}<br />
}<br />
</nowiki></div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Omega(N)&diff=2916
Omega(N)
2010-02-01T12:32:18Z
<p>Klasm: New page: == Definitions == In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> w...</p>
<hr />
<div>== Definitions ==<br />
<br />
In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math></math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.<br />
<br />
<br />
Here [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5593] Kevin O'Bryant gave an expression for <math> \omega(N)</math> as the minimum value of a rational function.<br />
<br />
<math>\omega(N)</math> is the infimum of an easily expressed rational expression with real variables.<br />
<br />
To wit, we want to handle<br />
<br />
<math> \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2</math><br />
<br />
where <math> g</math> is multiplicative, i.e., <math>g(mn)=g(m)g(n)</math>, and <math> g</math> takes values in the complex unit circle.<br />
<br />
Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:<br />
<br />
<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-j) \cos(2\pi (x_i - x_j)) \right)</math><br />
<br />
Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle<br />
<br />
<math> \omega(N,g) =\frac1N \sum_{i,j}^N \frac{(n-j+1) \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math><br />
<br />
where the multiplicativity of <math> g</math> is expressed through<br />
<br />
<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}</math><br />
<br />
<br />
== Bound for <math> \omega(N)</math> ==<br />
<br />
Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.<br />
<br />
For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}<br />
<br />
<nowiki><br />
{<br />
{0.5, {y[2] -> -1.462885206912493*^8}}, {0.4999999999999996, <br />
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, <br />
{0.4999999999999998, {y[2] -> -3.910648170382393*^8, <br />
y[3] -> 7.310011357069776*^8}}, {0.4801338227148131, <br />
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, <br />
y[5] -> 1.0297311434019392}}, {0.4834785819233648, <br />
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, <br />
y[5] -> -2.482521195509378}}, {0.4438473047036603, <br />
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927, <br />
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}}, <br />
{0.44836058320767935, {y[2] -> -6.302328081648096, <br />
y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114, <br />
y[7] -> 1.3714086295555794}}, {0.4390526818059849, <br />
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742, <br />
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}}, <br />
{0.4482580279013127, {y[2] -> -11.150737464977489, <br />
y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386, <br />
y[7] -> 1.2937492597484772}}, {0.42183629104159226, <br />
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959, <br />
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078, <br />
y[11] -> -4.037681072308063}}, {0.46807426662150586, <br />
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146, <br />
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568, <br />
y[11] -> -0.1948912567743111}}, {0.4461504388077101, <br />
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542, <br />
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343, <br />
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}}, <br />
{0.440086847216118, {y[2] -> -8.444874723393042*^6, <br />
y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546, <br />
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497, <br />
y[13] -> -0.3860579396004424}}, {0.4511481156025372, <br />
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454, <br />
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439, <br />
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}}, <br />
{0.4879779444860755, {y[2] -> -1.9863015651557585, <br />
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723, <br />
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203, <br />
y[13] -> 8.745313251473577}}, {0.49055456403195025, <br />
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378, <br />
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463, <br />
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*^6, <br />
y[17] -> -1.3074794690981901}}, {0.4762504985563837, <br />
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966, <br />
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052, <br />
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128, <br />
y[17] -> -0.5078441165548871}}, {0.4746150602095098, <br />
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034, <br />
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055, <br />
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*^7, <br />
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}}, <br />
{0.46698943639051865, {y[2] -> 2.1371340049709624, <br />
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044, <br />
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319, <br />
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675, <br />
y[19] -> 3.324056147899286}}, {0.4797097621551765, <br />
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416, <br />
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224, <br />
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*^7, <br />
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}}, <br />
{0.5132338128790291, {y[2] -> -35.83900453910118, <br />
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513, <br />
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877, <br />
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155, <br />
y[19] -> 0.26248549356388395}}, {0.5153366188102717, <br />
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087, <br />
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166, <br />
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045, <br />
y[17] -> 2.7485136995599563*^6, y[19] -> -0.043511090433659166, <br />
y[23] -> -0.4227059718591429}}, {0.507702455139519, <br />
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156, <br />
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231, <br />
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391, <br />
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828, <br />
y[23] -> -4.976865996291514*^6}}, {0.5096569558604533, <br />
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325, <br />
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692, <br />
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017, <br />
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205, <br />
y[23] -> -6.566392726129779}}, {0.4957655519078268, <br />
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567, <br />
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436, <br />
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648, <br />
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665, <br />
y[23] -> -10.457585568248941}}, {0.493353344030963, <br />
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923, <br />
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685, <br />
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408, <br />
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656, <br />
y[23] -> -6.311369549512267}}, {0.5395105053136733, <br />
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344, <br />
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131, <br />
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886, <br />
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113, <br />
y[23] -> -3.043577299897147}}, {0.5263782857581697, <br />
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909, <br />
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098, <br />
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125, <br />
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348, <br />
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}}, <br />
{0.532262482824738, {y[2] -> 8.357384007978142, <br />
y[3] -> 2.048744093019726*^7, y[5] -> 1.6341728378280016, <br />
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976, <br />
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193, <br />
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875, <br />
y[29] -> -0.06986050043371135}}, {0.5098368447944086, <br />
{y[2] -> -4.295234386315433*^7, y[3] -> -3.776498062431739, <br />
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955, <br />
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002, <br />
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923, <br />
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647, <br />
y[31] -> -0.028361936941826257}}, {0.5079375830113259, <br />
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114, <br />
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977, <br />
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327, <br />
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595, <br />
y[23] -> -6.595269670599772*^6, y[29] -> 1.2322052779098513, <br />
y[31] -> 0.21494585293809093}}, {0.49555821568780584, <br />
{y[2] -> -5.066246086172957*^6, y[3] -> -3.630034375104794, <br />
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798, <br />
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077, <br />
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664, <br />
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956, <br />
y[31] -> 0.032379893238005227}}, {0.540002424826123, <br />
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504, <br />
y[5] -> 2.4225107795045388*^8, y[7] -> 0.04290314638959777, <br />
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438, <br />
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788, <br />
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*^7, <br />
y[31] -> -0.8641525290683024}}, {0.5374015273420782, <br />
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875, <br />
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146, <br />
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739, <br />
y[17] -> -5.558626478659901*^6, y[19] -> 0.4653174372715049, <br />
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646, <br />
y[31] -> 0.7701814129443211}}, {0.5538027559050895, <br />
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013, <br />
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657, <br />
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118, <br />
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456, <br />
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132, <br />
y[31] -> 0.8694448889517893}}, {0.5410949110097839, <br />
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534, <br />
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567, <br />
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142, <br />
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896, <br />
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696, <br />
y[31] -> -4.992562635912787*^6, y[37] -> -0.7571633316839521}}, <br />
{0.5429973547947488, {y[2] -> -6.844959654454303, <br />
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003, <br />
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883, <br />
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857, <br />
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862, <br />
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066, <br />
y[37] -> -0.002931194578079365}}, {0.5467999587431224, <br />
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493, <br />
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871, <br />
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538, <br />
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862, <br />
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208, <br />
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}}, <br />
{0.5559476548255569, {y[2] -> -9.848512412603167, <br />
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561, <br />
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056, <br />
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*^6, <br />
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, <br />
y[29] -> 1.2680941513401158*^7, y[31] -> 0.5666075600978651, <br />
y[37] -> -0.0937132554032237}}, {0.5505709016488777, <br />
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795, <br />
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243, <br />
y[11] -> -3.47473672784074*^6, y[13] -> -0.6567175708714164, <br />
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*^6, <br />
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, <br />
y[31] -> -1.9302319705248326*^6, y[37] -> 4.090882786292679, <br />
y[41] -> -0.5196037223519064}}, {0.5410183504459096, <br />
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326, <br />
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513, <br />
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123, <br />
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914, <br />
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003, <br />
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513, <br />
y[41] -> 0.8078091963254634}}, {0.5419769095397796, <br />
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858, <br />
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796, <br />
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516, <br />
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593, <br />
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716, <br />
y[31] -> 5.116561470832145*^6, y[37] -> 3.090373695813483, <br />
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}}, <br />
{0.5447128396652148, {y[2] -> 3.5104534047833686, <br />
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708, <br />
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171, <br />
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481, <br />
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323, <br />
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*^6, <br />
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384, <br />
y[43] -> 4.143328442778921}}, {0.5621737511465918, <br />
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912, <br />
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136, <br />
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851, <br />
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522, <br />
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176, <br />
y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, <br />
y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, <br />
{0.5458049553223693, {y[2] -> -7.626649842911723, <br />
y[3] -> 3.182773873173586*^7, y[5] -> -1.9670364421721978, <br />
y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, <br />
y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, <br />
y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, <br />
y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, <br />
y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, <br />
y[43] -> 0.5802847144626384}}, {0.5538527358037998, <br />
{y[2] -> -6.019028440570796*^6, y[3] -> 3.9774063888198805, <br />
y[5] -> 1.627686965541912*^6, y[7] -> 0.1876670149531972, <br />
y[11] -> -6.627729461561532*^6, y[13] -> 0.45376652495185976, <br />
y[17] -> -2.839219113548976, y[19] -> -0.11532275380682605, <br />
y[23] -> 6.7198495482116405, y[29] -> -3.467411847670985, <br />
y[31] -> -0.07037035912929553, y[37] -> 0.591684721532398, <br />
y[41] -> 5.777047972367371*^6, y[43] -> -0.07507844950087077, <br />
y[47] -> -4.358887509906755}}, {0.5401308818895296, <br />
{y[2] -> 3.389795734987173, y[3] -> 3.8668146046647127, <br />
y[5] -> 0.8577559069300346, y[7] -> -1.6295597106276374, <br />
y[11] -> 0.5231413760313481, y[13] -> 742.3888224076079, <br />
y[17] -> -0.24443014877037325, y[19] -> -4.201569449912788, <br />
y[23] -> 0.19884266463491398, y[29] -> -0.4371033927898226, <br />
y[31] -> -246.0221156311323, y[37] -> 1.8249044545155355, <br />
y[41] -> 0.0988860127802216, y[43] -> -384.1591004952399, <br />
y[47] -> -1.7538440569960194}}, {0.5363103521139646, <br />
{y[2] -> 3.333020223531083, y[3] -> 3.7429353161140133, <br />
y[5] -> 0.8786201463515133, y[7] -> -1.5389392753979991, <br />
y[11] -> 0.5425974766008388, y[13] -> 1.3323084091161266*^7, <br />
y[17] -> -0.24578494270446172, y[19] -> -3.8771464040114823, <br />
y[23] -> 0.2244303634237275, y[29] -> -0.4373275283960972, <br />
y[31] -> -632384.655380349, y[37] -> 1.9488891017929366, <br />
y[41] -> 0.0887064268815982, y[43] -> -8.803504041720465*^6, <br />
y[47] -> -1.3892438499060007}}, {0.5453879839837351, <br />
{y[2] -> 2.73992352356985, y[3] -> 2.817657434467374, <br />
y[5] -> 0.6692134878352286, y[7] -> -1.9140977785470707, <br />
y[11] -> 0.35729190928335297, y[13] -> 20.108997104450143, <br />
y[17] -> -0.42503771960213765, y[19] -> 1.0154323728067767*^6, <br />
y[23] -> -0.12940157913690162, y[29] -> -0.8650874405339943, <br />
y[31] -> 3.1668834978424774, y[37] -> 1.1756543193093625, <br />
y[41] -> -0.13564431449224323, y[43] -> -1.548989386225219*^6, <br />
y[47] -> 1.3757810922960734*^7}}, {0.544623917879015, <br />
{y[2] -> -1.6079312057861998*^6, y[3] -> 3.818710615864944, <br />
y[5] -> 2.0416373983878603*^6, y[7] -> 0.0796985220985775, <br />
y[11] -> 20.65800889056641, y[13] -> 0.3991585241146419, <br />
y[17] -> -3.3817561255470014, y[19] -> -0.11686135066393119, <br />
y[23] -> 1.1765289825982254*^6, y[29] -> -4.829380102405632, <br />
y[31] -> -0.03990492245008383, y[37] -> 0.4829417569781351, <br />
y[41] -> 4.097142033699167, y[43] -> -0.3866830584564186, <br />
y[47] -> -1.0493377612540264}}, {0.5574945212289636, <br />
{y[2] -> 3.521815923555302, y[3] -> -1.1058194115348723, <br />
y[5] -> 1.745781769078708, y[7] -> -0.8045262105949067, <br />
y[11] -> 0.7755941102831237, y[13] -> -0.9206605312412145, <br />
y[17] -> 0.8588317087691079, y[19] -> -3.4018962919312776, <br />
y[23] -> 0.46293383131411336, y[29] -> -0.3339405953430995, <br />
y[31] -> -2.912189967123192*^6, y[37] -> -1.7422201839944014, <br />
y[41] -> 0.6419564629044578, y[43] -> 7.1380131906509465, <br />
y[47] -> 0.19419997803249356}}, {0.5568512079607693, <br />
{y[2] -> -1847.0162162204433, y[3] -> 3.7433300197926243, <br />
y[5] -> 6411.295532189543, y[7] -> 0.08495666275018456, <br />
y[11] -> -1998.15565755151, y[13] -> 0.40879440723628807, <br />
y[17] -> -3.8984872437145044, y[19] -> -0.18293043611329182, <br />
y[23] -> 10.530816850822847, y[29] -> -4.489590552645085, <br />
y[31] -> -0.13194590199381787, y[37] -> 0.5263403371559953, <br />
y[41] -> 5.49582466392191, y[43] -> -0.30601196489854615, <br />
y[47] -> -1.3582722412479475, y[53] -> -2.8066662056917093}}, <br />
{0.559983380057825, {y[2] -> -1.3388995684634393, <br />
y[3] -> 5.218739739233073, y[5] -> -0.2047600422912967, <br />
y[7] -> 1.9797665592396885*^7, y[11] -> 1.3887034097281536, <br />
y[13] -> -2.0612517427474972, y[17] -> 0.8862869327940438, <br />
y[19] -> -0.3098254211547697, y[23] -> -2.5901266177896416*^8, <br />
y[29] -> -2.2525950436074313, y[31] -> 0.6084446735574678, <br />
y[37] -> -0.39854564144349475, y[41] -> 2.7526079883448524, <br />
y[43] -> 0.6874792732607808, y[47] -> -3.764709481763189, <br />
y[53] -> 3.564174528271919}}, {0.5504879009429681, <br />
{y[2] -> 9.03123955295885, y[3] -> 2.2136609253188295, <br />
y[5] -> -2.0678258624188232*^6, y[7] -> -0.09413658614786298, <br />
y[11] -> 5.717366741226189, y[13] -> 0.15567419778859554, <br />
y[17] -> -7.208623752944289*^6, y[19] -> -0.6977550889277575, <br />
y[23] -> 2.0936489384289856, y[29] -> -2.7605129103186652*^7, <br />
y[31] -> -0.5513912542393178, y[37] -> 0.10221458100687682, <br />
y[41] -> 2.1640423991444195, y[43] -> -0.5875636480502329, <br />
y[47] -> 1.715499959928509*^7, y[53] -> 2.6616582219247134}}, <br />
{0.5458851545432952, {y[2] -> -5.287161241095462, <br />
y[3] -> -1.6583162153081066, y[5] -> -3.3288334101794765, <br />
y[7] -> 0.40288657194056393, y[11] -> -1.5010982738523997, <br />
y[13] -> 0.2588903684631928, y[17] -> -3.436013244363909, <br />
y[19] -> 1.1334615150683407, y[23] -> -0.9518246993073657, <br />
y[29] -> -1.7830036107943674, y[31] -> 1.3014159974388895, <br />
y[37] -> 0.6196183103799945, y[41] -> -0.17653284626235025, <br />
y[43] -> -8.514765322324852*^6, y[47] -> -4.189583302785, <br />
y[53] -> -0.7131516640451642}}, {0.5491779017342883, <br />
{y[2] -> -2.3847076799217017, y[3] -> 1.4044818863482407, <br />
y[5] -> -0.9625963185392208, y[7] -> 1.7900975653441773, <br />
y[11] -> -0.06465498141384049, y[13] -> 4.817235039864703, <br />
y[17] -> -0.25310986318020484, y[19] -> -4.970483414380786, <br />
y[23] -> 0.4486821965367979, y[29] -> 1.091907070957466, <br />
y[31] -> -1.063528493284542, y[37] -> 171810.1754353715, <br />
y[41] -> -0.024619668315839376, y[43] -> -0.9827488606119602, <br />
y[47] -> 1.1619366272362106, y[53] -> 0.41065985901822705}}, <br />
{0.5605581749105492, {y[2] -> 3.027632580117612, <br />
y[3] -> 3.1960336822744946, y[5] -> 0.7859347217908186, <br />
y[7] -> -1.6075367090588815, y[11] -> 0.420302061789829, <br />
y[13] -> 2.834085806987071*^7, y[17] -> -0.39511691032585206, <br />
y[19] -> 1.2560796536845084*^7, y[23] -> 0.0318390685791732, <br />
y[29] -> -0.5333456813624734, y[31] -> -3.473422724807593*^7, <br />
y[37] -> 1.6556532654346816, y[41] -> 0.1096024123215574, <br />
y[43] -> -5.778600356830014*^6, y[47] -> -1.6651426336543034, <br />
y[53] -> -0.5725276311066366}}, {0.5524591153632439, <br />
{y[2] -> 9.15237409161851, y[3] -> 2.009327756964571, <br />
y[5] -> 743539.9436168781, y[7] -> -0.08466691902541673, <br />
y[11] -> 5.817481302444094, y[13] -> 0.1675306932760601, <br />
y[17] -> 2.0776343830107022*^6, y[19] -> -0.7226602411236893, <br />
y[23] -> 2.1893209174225303, y[29] -> 3.880714074279567, <br />
y[31] -> -0.7068785155011873, y[37] -> -0.034544725241448075, <br />
y[41] -> 1.7753351439100655, y[43] -> -0.6472691827015548, <br />
y[47] -> 368510.4969259753, y[53] -> -582724.4120164809, <br />
y[59] -> 311442.9510525222}}, {0.5606497733587336, <br />
{y[2] -> 3.3299236312177247, y[3] -> 4.242923677984408, <br />
y[5] -> 0.9732763760852335, y[7] -> -1.1834066132028593, <br />
y[11] -> 0.7126644769358326, y[13] -> 1.3956387099741682*^7, <br />
y[17] -> -0.2429854139189772, y[19] -> -1.9477415056452971, <br />
y[23] -> 0.37600159256085175, y[29] -> -0.12996087369871648, <br />
y[31] -> 546418.5231248025, y[37] -> -1.2261669343406128*^7, <br />
y[41] -> 0.2107116603678402, y[43] -> -5.255145411844228*^6, <br />
y[47] -> -0.9891264323012026, y[53] -> -0.36952794256418503, <br />
y[59] -> -0.629177500336727}}, {0.5409409222767462, <br />
{y[2] -> -9.386040329445352, y[3] -> -1.9982212173963287, <br />
y[5] -> 51868.562388185535, y[7] -> 0.11485671711460184, <br />
y[11] -> -4.152230885212978, y[13] -> -0.10985151504318237, <br />
y[17] -> -12685.789320977634, y[19] -> 0.7939454612005391, <br />
y[23] -> -2.090382110891867, y[29] -> -3.8368801260586185, <br />
y[31] -> 0.62666251421115, y[37] -> 0.09888542996449518, <br />
y[41] -> -2.0490693623989022, y[43] -> 0.6618361754535906, <br />
y[47] -> -4837.19011514421, y[53] -> -4.622966232274686, <br />
y[59] -> -14.441877571818774, y[61] -> -1.0854855882356003}}, <br />
{0.5504930357833084, {y[2] -> -1.5684229702045942, <br />
y[3] -> 2.1906487684885048, y[5] -> -0.536467870283022, <br />
y[7] -> 6.32778369271226, y[11] -> 0.5034630491164894, <br />
y[13] -> -4.401789242341788, y[17] -> 0.40094213785808164, <br />
y[19] -> -1.1615966004682858, y[23] -> 1.5792591091888084, <br />
y[29] -> -6.826050191930487*^6, y[31] -> 0.09419851884862504, <br />
y[37] -> -1.4087925782385866, y[41] -> 0.8014115278033986, <br />
y[43] -> 0.02574018267373435, y[47] -> -6.10698431819938, <br />
y[53] -> 2.408444849978389, y[59] -> -1.3987046103733207, <br />
y[61] -> 0.4969984166077462}}, {0.5446604430297907, <br />
{y[2] -> -1.6220448040430115, y[3] -> 2.054026001427892, <br />
y[5] -> -0.571844543206896, y[7] -> 5.300025349396089, <br />
y[11] -> 0.4404535557144669, y[13] -> -5.207974606119084, <br />
y[17] -> 0.33504781866094996, y[19] -> -1.298493369878154, <br />
y[23] -> 1.4502910251112735, y[29] -> 7.069226385314122, <br />
y[31] -> -0.048814751106247355, y[37] -> -1.6559183485457014, <br />
y[41] -> 0.7030881474104027, y[43] -> -0.05489963510807713, <br />
y[47] -> -8.73066370716909, y[53] -> 2.0902418324716434, <br />
y[59] -> -2.0900648774190254, y[61] -> 0.3115712423996155}}, <br />
{0.5485208904992191, {y[2] -> -9.83702338729878, <br />
y[3] -> -1.9961248696515557, y[5] -> 1.4149705746421264*^7, <br />
y[7] -> 0.11182687628643254, y[11] -> -4.193101970653638, <br />
y[13] -> -0.11080855361674405, y[17] -> -2.005889406534506*^8, <br />
y[19] -> 0.784117812481433, y[23] -> -2.174515779870723, <br />
y[29] -> -3.6240125408312136, y[31] -> 0.62017956738753, <br />
y[37] -> 0.09673506020361744, y[41] -> -2.1525947194438397, <br />
y[43] -> 0.6531445024561217, y[47] -> -5833.422440596358, <br />
y[53] -> -4.822732802784665, y[59] -> -1.430764520502954*^6, <br />
y[61] -> -0.6400367372252429}}, {0.5637147347732624, <br />
{y[2] -> -15.6818734164482, y[3] -> -2.1326753043276208, <br />
y[5] -> 184493.3758547012, y[7] -> 0.08676796846893833, <br />
y[11] -> -3.9426557111054668, y[13] -> -0.06173668229848903, <br />
y[17] -> -3.4718205438529723*^6, y[19] -> 0.7277241103464438, <br />
y[23] -> -2.6056094534024252, y[29] -> -5.088663679678725, <br />
y[31] -> 0.4896481276398145, y[37] -> -0.09872963368027905, <br />
y[41] -> -2.0689071131284584*^6, y[43] -> 0.3967585631083156, <br />
y[47] -> -4532.596733187516, y[53] -> -2.990463032043736, <br />
y[59] -> -1250.0646409875155, y[61] -> -0.6514894080877749}}, <br />
{0.5574971466785472, {y[2] -> -12.89831349714744, <br />
y[3] -> -1.989556552930725, y[5] -> 4.1124340521452166*^6, <br />
y[7] -> 0.10572842024786352, y[11] -> -3.4077847397096166, <br />
y[13] -> -0.050718117941117684, y[17] -> -2.334626139952064*^6, <br />
y[19] -> 0.8159882179135113, y[23] -> -2.2112753444437456, <br />
y[29] -> -3.687956931286009, y[31] -> 0.5459441985044045, <br />
y[37] -> 0.07470589842936966, y[41] -> -3.783770929541928, <br />
y[43] -> 0.5709022698524843, y[47] -> -6.740703296130651*^8, <br />
y[53] -> -2.917504195233325, y[59] -> -10.338760065079597, <br />
y[61] -> -0.4706281888552907}}, {0.5512236848530143, <br />
{y[2] -> 2.678172582377694, y[3] -> -1.310374883095014, <br />
y[5] -> 0.9939841059079616, y[7] -> -1.7785615036215077, <br />
y[11] -> 0.0027003343101931385, y[13] -> 648987.24460795, <br />
y[17] -> 0.21487666688255183, y[19] -> 4.660126311569545, <br />
y[23] -> -0.6318210504304709, y[29] -> -0.5127738485630606, <br />
y[31] -> 1.5564432856444592, y[37] -> -5.79749937222906, <br />
y[41] -> 0.20003521857360163, y[43] -> 1.2395292870364005, <br />
y[47] -> -1.1826243887259602, y[53] -> -0.7554766844701083, <br />
y[59] -> -1.9012403142168892, y[61] -> 1.173588547371486, <br />
y[67] -> -5.229169044686542}}, {0.5499121940594726, <br />
{y[2] -> 2.760226848334816, y[3] -> -1.2804958043686288, <br />
y[5] -> 1.0219649911436968, y[7] -> -1.7039159743515304, <br />
y[11] -> 0.02219347957585562, y[13] -> 322195.1713625446, <br />
y[17] -> 0.2434544060712443, y[19] -> 5.496764817454516, <br />
y[23] -> -0.5877510363659428, y[29] -> -0.46329863229437623, <br />
y[31] -> 1.6963812464375045, y[37] -> -4.599583164536985, <br />
y[41] -> 0.25787524309930243, y[43] -> 1.3686927852936006, <br />
y[47] -> -1.0630953287441995, y[53] -> -0.7311504933506725, <br />
y[59] -> -1.7494697067352272, y[61] -> 1.2561845523605812, <br />
y[67] -> -182102.170199389}}, {0.5483080826469182, <br />
{y[2] -> 2.6498364464076207, y[3] -> 3.021126152636368, <br />
y[5] -> 0.7572348577185984, y[7] -> -1.700084208200631, <br />
y[11] -> 0.43625796402857475, y[13] -> -46971.63170860386, <br />
y[17] -> -0.3674272230009722, y[19] -> 78070.09472073127, <br />
y[23] -> -0.0982653546810197, y[29] -> -0.5910759281137994, <br />
y[31] -> 5.286514869512648, y[37] -> 1.537726184125387, <br />
y[41] -> -0.01563127769541324, y[43] -> -48660.466404012375, <br />
y[47] -> -2.2621133513794702, y[53] -> -0.44505813880766654, <br />
y[59] -> -1.4966875853517387, y[61] -> 0.2618818317244051, <br />
y[67] -> 0.6209965718708607}}, {0.5451386267895824, <br />
{y[2] -> 8.403675409897469, y[3] -> -10.104394213837576, <br />
y[5] -> 2.698078123592497, y[7] -> -0.5394067302483168, <br />
y[11] -> 3.075874159362834, y[13] -> -1.0441560792891287, <br />
y[17] -> 1.1169290721098575, y[19] -> -0.30821564905740756, <br />
y[23] -> 2.637469168561796, y[29] -> 1.0015370671088775, <br />
y[31] -> -0.61291012605091, y[37] -> -1.7496544526517577, <br />
y[41] -> 1.842606432699937, y[43] -> -0.3452640617813726, <br />
y[47] -> 0.15988379210506262, y[53] -> 1.3729270793222859, <br />
y[59] -> 0.6241629125512985, y[61] -> 96289.16955508286, <br />
y[67] -> -2.3810259314590665}}, {0.5344272945207329, <br />
{y[2] -> 165.10277230321324, y[3] -> -3.308487268972249, <br />
y[5] -> 12.455893515741943, y[7] -> -0.1570711195130275, <br />
y[11] -> -28.07211028229915, y[13] -> -0.37078848866130276, <br />
y[17] -> 4.029899110329946, y[19] -> 0.24909973367051877, <br />
y[23] -> -12.88646722015868, y[29] -> 5.519025878986118, <br />
y[31] -> 0.07853281950302134, y[37] -> -0.5099664379517453, <br />
y[41] -> 558668.190555858, y[43] -> 0.2585862768213708, <br />
y[47] -> 1.1914196990609796, y[53] -> -100714.3994553162, <br />
y[59] -> 2.8162193059622065, y[61] -> -1.2347666983734868, <br />
y[67] -> -0.4544987489160042, y[71] -> 105179.33224334147}}<br />
}<br />
</nowiki></div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Experimental_results&diff=2915
Experimental results
2010-02-01T12:06:47Z
<p>Klasm: </p>
<hr />
<div>[[The Erd&#337;s discrepancy problem|To return to the main Polymath5 page, click here]].<br />
<br />
Perhaps we should have two kinds of subpages to this page: Pages about finding examples, and pages about analyzing them?<br />
<br />
== Experimental data==<br />
* [[The first 1124-sequence]] with discrepancy 2. ''Some more description''<br />
* Other [[length 1124 sequences]] with discrepancy 2. ''Some more description''<br />
* A [[sequence of length 1091]] with discrepancy 2.<br />
* A [[sequence of length 1112]] derived from one with nice multiplicative properties.<br />
* [[Sequences given by modulated Sturmian functions]].<br />
* Some data about the problem with [[different upper and lower bound]]. Let N(a,b) be the largest N such that there is a sequence <math>x_1,\dots,x_N</math> all of whose HAP-errors are between -a and b, inclusive.<br />
* Sequences taking values in <math>\mathbb{T}</math>:<br />
** [[4th roots of unity]]<br />
** [[6th roots of unity]]<br />
* [http://thomas1111.wordpress.com/2010/01/10/tables-for-a-c10-candidate/ A sequence of length 407] with discrepancy 2 such that <math>x_n=x_{32 n}</math> for every n. [[The HAP-subsequence structure of that sequence]].<br />
* More [[T32-invariant sequences]].<br />
* Long [[multiplicative sequences]].<br />
* Long sequences satisfying [[T2(x) = -x]].<br />
* Long sequences satisfying [[T2(x) = T5(x) = -x]]<br />
* Long sequences satisfying [[T2(x) = -T3(x)]].<br />
* Long sequences satisfying constraints of the form [[T_m(x) = (+/-)T_n(x)]].<br />
* Table of [[longest constrained sequences]].<br />
* Table of [[short sequences statistics]].<br />
* [[Dirichlet inverses]] of good sequences.<br />
* Sequences with [[bounded Dirichlet inverse]].<br />
* [[Shifts and signs]] related to an interesting structure of the first 1124-sequence.<br />
* Long sequences with [[low discrepancy on PAPs]].<br />
* [[Forced Drifts in Multiplicative Sequences]]. The tree of possible assignments to primes is traversed, and for each possible assignment an interval is found where the function must have substantive drift. Data includes drifts of 2, 3, 4, and 5.<br />
*Bounds for the numbers [[Omega(N)]]<br />
<br />
<br />
<br />
==Source code==<br />
<br />
* [[Convert raw input string into CSV table]]<br />
* [[Create tables in an HTML file from an input sequence]]<br />
* [[Verify the bounded discrepancy of an input sequence]]<br />
* [[Depth-first search]]<br />
* [[Search for completely multiplicative sequences]]<br />
* [[Refined greedy computation of multiplicative sequences]]<br />
* [[Computing a HAP basis]]<br />
* [[Estimate the number of discrepancy 2 sequences]]<br />
* [[Updating partial sums with Fenwick tree]]<br />
<br />
==Wish list==<br />
<br />
There is a separate page for [[proposals for finding long low-discrepancy sequences]]. It goes without saying that implementing any of these proposals belongs to the wish list.<br />
<br />
<!--<br />
* What is the discrepancy of the sequence defined in [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/ this post], <br />
DONE, i think.<br />
--><br />
* Find long/longest quasi-multiplicative sequences with some fixed group G, function <math>G\to \{-1,1\}</math> and maximal discrepancy C<br />
** <math>G=C_6</math> and the function that sends 0,1 and 2 to 1 (because this seems to be a good choice)<br />
* Do a "Mark-Bennet-style analysis" of one of the new 1124-sequences. [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4827] Also [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4842 done] (by Mark Bennet).<br />
*. Take a moderately large k and search for the longest sequence of discrepancy 2 that's constructed as follows. First, pick a completely multiplicative function f to the group <math>C_{2k}</math>. Then set <math>x_n</math> to be 1 if f(n) lies between 0 and k-1, and -1 if f(n) lies between k and 2k-1. Alec has already [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4563 done this for k=1] and [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4734 partially done it for k=3].<br />
*Search for the longest sequence of discrepancy 2 with the property that <math>x_n=x_{32n}</math> for every n. The motivation for this is to produce a fundamentally different class of examples (different because their group structure would include an element of order 5). It's not clear that it will work, since 32 is a fairly large number. However, if you've chosen <math>x_{32n}</math> then that will have some influence on several other choices, such as <math>x_{4n},x_{8n}</math> and <math>x_{16n}</math>, so maybe it will lead to something interesting. Alec [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4873 has made a start on this] and an [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4874 initial investigation] suggests that the sequence he has found does indeed have some <math>C_{10}</math>-related structure. <br />
*Here's another experiment that should be pretty easy to program and might yield something interesting. It's to look at the how the discrepancy appears to grow when you define a sequence using a greedy algorithm. I say "a" greedy algorithm because there are various algorithms that could reasonably be described as greedy. Here are a few.<br />
<br />
<blockquote><br />
1. For each n let <math>x_n</math> be chosen so as to minimize the discrepancy so far, given the choices already made for <math>x_1,\dots,x_{n-1}</math>. (If this does not uniquely determine <math>x_n</math> then choose it arbitrarily, or randomly, or according to some simple rule like always equalling 1.)<br />
</blockquote><br />
<br />
<blockquote><br />
2. Same as 1 but with additional constraints, in the hope that these make the sequence more likely to be good. For instance, one might insist that <math>x_{2k}=x_{3k}</math> for every k. Here, when choosing <math>x_n</math> one would probably want to minimize the discrepancy up to <math>x_{n+k}</math> if <math>x_{n+1},\dots,x_{n+k}</math> had already been chosen. Another obvious constraint to try is complete multiplicativity.<br />
</blockquote><br />
<br />
<blockquote><br />
3. A greedy algorithm of sorts, but this time trying to minimize a different parameter. The first algorithm will do this: when you pick <math>x_n</math> you look, for each factor d of n, at the partial sum along the multiples of d up to but not including n. This will give you a set A of numbers (the possible partial sums). If max(A) is greater than max(-A) then you set <math>x_n=-1</math>, if max(-A) is greater than max(A) then you let <math>x_n=1</math>, and if they are equal then you make the decision according to some rule that seems sensible. But it might be that you would end up with a slower-growing discrepancy if you regarded A as a multiset and made the decision on some other basis. For instance, you could take the sum of <math>2^k</math> over all positive elements <math>k\in A</math> (with multiplicity) and the sum of <math>2^{-k}</math> over all negative elements and choose <math>x_n</math> according to which was bigger. Although that wouldn't minimize the discrepancy at each stage, it might make the sequence better for future development because it wouldn't sacrifice the needs of an overwhelming majority to those of a few rogue elements.<br />
</blockquote><br />
<br />
<blockquote><br />
4. A greedy algorithm to choose a good completely multiplicative low-discrepancy sequence. Now you are free only to choose the values at primes. If you have chosen the values up to but not including p, then fill in all the values that are forced by multiplicativity and then make whatever seems to be the best choice for the value at p. Again, there are several approaches that could be reasonable here. One is simply to ensure that the partial sum of the sequence up to p is as small (in modulus) as you can make it. But that would be foolish if you've already filled in the values at p+1,...,p+k. So an only slightly less greedy algorithm is to look at the effect of your choice at p on the partial sums all the way up to the next prime and choose the best value accordingly. If you do that, then at what rate do the partial sums grow? In particular, do they grow at least logarithmically? [http://michaelnielsen.org/polymath1/index.php?title=Multiplicative_sequences#Minimizing_D_up_to_the_next_prime This is being addressed here]<br />
</blockquote><br />
<br />
<blockquote><br />
The motivation for these experiments is to see whether they, or some variants, appear to lead to sublogarithmic growth. If they do, then we could start trying to prove rigorously that sublogarithmic growth is possible. I still think that a function that arises in nature and satisfies f(1124)=2 ought to be sublogarithmic.<br />
</blockquote><br />
<br />
*What happens if one applies a backtracking algorithm to try to extend the following discrepancy-2 sequence, which satisfies <math>x_{2n}=-x_n</math> for every n, to a much longer discrepancy-2 sequence: + - - + - + + - - + + - + - + + - + - - + - - + + - - + + - + - - + - - + + + + - - - + + + - - + - + + - + - - + - ? This question has been answered [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4893 in the comments following the asking of the question on the blog]. <br />
<br />
* Investigate what happens if our HAPs are restricted to allow differences divisible only by 2 or 3 [and then other sets of primes including 2] - {2,3,5,7} would be interesting - is there an infinite sequence of discrepancy 2 in these simple cases - is it easy to find an infinite sequence with finite discrepancy in these cases? [for sets of odd primes, take a sequence which is 1 on odd numbers, -1 on even numbers. Including 2 is the non-trivial case]. It is possible that completely multiplicative sequences could be found for some of these cases.<br />
<br />
* Compute the Dirichlet series <math>f(s) = \sum x_n n^{-s}</math> for some of our long low-discrepancy series, and see what this function looks like in the vicinity of <math>1</math>, and elsewhere. [http://gowers.wordpress.com/2010/01/11/the-erds-discrepancy-problem-iii/#comment-5062 Alec has now looked at this].<br />
<br />
*Take a long sequence <math>(x_n)</math> of discrepancy 2 and try to create a new long sequence <math>(y_n)</math> subject to the constraint that <math>y_{2n}=x_n</math>. How far does one typically get before getting stuck? And how much further does one get if one uses the resulting sequence as a seed for the usual algorithm? [http://gowers.wordpress.com/2010/01/14/the-erds-discrepancy-problem-iv/#comment-5096 One does not get too far, as Alec showed].<br />
<br />
*Take some of the good sequences and calculate the following parameter, which is supposed to measure the amount of multiplicity. If the sequence is defined up to n, then the parameter is the expected value of <math>x_ax_bx_cx_d</math> over all quadruples (a,b,c,d) such that a,b,c,d are at most n and ab=cd. I'm expecting that the answer will be significantly greater than zero -- perhaps something like 0.3 -- but would like to have this confirmed. [http://gowers.wordpress.com/2010/01/14/the-erds-discrepancy-problem-iv/#comment-5132 It has now been confirmed].<br />
<br />
* ... you are welcome to add more.</div>
Klasm
https://michaelnielsen.org/polymath/index.php?title=Talk:Main_Page&diff=1757
Talk:Main Page
2009-06-24T13:08:48Z
<p>Klasm: /* Spam and disabling anonymous editing */</p>
<hr />
<div>== Spam and disabling anonymous editing ==<br />
<br />
Spam seems to be quite a problem. Most (all?) of the spam is coming from anonymous accounts. What would other people think about changing the wiki settings so only people logged in can make edits? On many wikis this greatly reduces (but does not necessarily entirely eliminate) spam. If I get a couple of affirmative responses from active editors, I'll make the change.<br />
--[[User:WikiSysop|WikiSysop]] 11:13, 24 June 2009 (UTC) (Michael Nielsen)<br />
<br />
<br />
I would be in favour of requiring log in. I have deleted spam entries and restored entries overwritten by spam more or less daily. (Klas Markström)<br />
<br />
== Polymath Logo ==<br />
<br />
It would be cool to have a polymath logo to replace "set $wgLogo to the URL path...".</div>
Klasm