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:<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> | :<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> | ||
== Exact values == | |||
These are the currently known values of <math>\Omega(N)</math> | |||
<math>\Omega(1)=1</math> at <math>y_1=0</math> | |||
<math>\Omega(2)=\frac{1}{2}</math> at <math>y_1=0, y_2=-\infty</math> | |||
<math>\Omega(3)=\frac{1}{2}</math> at <math>y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}}</math> | |||
<math>\Omega(4)=\frac{1}{2}</math> | |||
== Upper bounds for <math> \omega(N)</math> == | == Upper bounds for <math> \omega(N)</math> == | ||
Revision as of 01:11, 2 February 2010
Definitions
In this [1] comment Terrence Tao defined a function [math]\displaystyle{ \omega(N) }[/math] which is a lower bound for the discrepancy of a completely multiplicative function. If [math]\displaystyle{ \omega(N) }[/math] is unbounded as a function of [math]\displaystyle{ N }[/math] then all compeltely mutplicative functions will have unbounded discrepancy.
Here [2] Kevin O'Bryant gave an expression for [math]\displaystyle{ \omega(N) }[/math] as the minimum value of a rational function.
[math]\displaystyle{ \omega(N) }[/math] is the infimum of an easily expressed rational expression with real variables.
To wit, we want to handle
[math]\displaystyle{ \omega(N,g) := \frac1N \sum_{n=1}^N \left|\sum_{k=1}^n g(k)\right|^2 }[/math]
where [math]\displaystyle{ g }[/math] is multiplicative, i.e., [math]\displaystyle{ g(mn)=g(m)g(n) }[/math], and [math]\displaystyle{ g }[/math] takes values in the complex unit circle.
Setting [math]\displaystyle{ g(k)=e^{2\pi i x_k} }[/math], the multiplicativity of [math]\displaystyle{ g }[/math] becomes [math]\displaystyle{ x_{mn}=x_m+x_n }[/math]. We wish to handle:
[math]\displaystyle{ \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]
Setting [math]\displaystyle{ y_k = \tan(\pi x_k) }[/math] (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle
[math]\displaystyle{ \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]
where the multiplicativity of [math]\displaystyle{ g }[/math] is expressed through
[math]\displaystyle{ y_{mn} = \frac{y_m+y_n}{1-y_my_n}. }[/math]
A variant of the above expression for [math]\displaystyle{ \omega(N) }[/math] is
- [math]\displaystyle{ \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]
Exact values
These are the currently known values of [math]\displaystyle{ \Omega(N) }[/math]
[math]\displaystyle{ \Omega(1)=1 }[/math] at [math]\displaystyle{ y_1=0 }[/math]
[math]\displaystyle{ \Omega(2)=\frac{1}{2} }[/math] at [math]\displaystyle{ y_1=0, y_2=-\infty }[/math]
[math]\displaystyle{ \Omega(3)=\frac{1}{2} }[/math] at [math]\displaystyle{ y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}} }[/math]
[math]\displaystyle{ \Omega(4)=\frac{1}{2} }[/math]
Upper bounds for [math]\displaystyle{ \omega(N) }[/math]
Using a numerical optimization routine the following upper bounds on [math]\displaystyle{ \omega(N) }[/math] were found.
For [math]\displaystyle{ 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]}}
A plot of the bounds for [math]\displaystyle{ N=1,...,84 }[/math] [3]
{ {1,
{y[1]->0}},
{1/2,
{y[2] -> -Infinity}},
{0.4999999999999996,
{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}},
{0.4999999999999998,
{y[2] -> -Infinity, y[3] -> Infinity}},
{0.4801338227148131,
{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}},
{0.4834785819233648,
{y[2] -> -9.507430209541221, y[3] -> 5.627112246360941, y[5] -> -2.482521195509378}},
{0.4438473047036603,
{y[2] -> -8.63996974968336, y[3] -> 4.836164095730927,
y[5] -> -1.7254441087345165, y[7] -> 2.6554597438188194}},
{0.44836058320767935,
{y[2] -> -6.302328081648096, y[3] -> 5.192573425223564, y[5] -> -1.5213118631968114,
y[7] -> 1.3714086295555794}},
{0.4390526818059849,
{y[2] -> -15.101875241283222, y[3] -> 8.7551999765742,
y[5] -> -2.3675892230191136, y[7] -> 0.979892667288003}},
{0.4482580279013127,
{y[2] -> -11.150737464977489, y[3] -> 5.2718510439642365, y[5] -> -1.931472713727386,
y[7] -> 1.2937492597484772}},
{0.42183629104159226,
{y[2] -> -12.020130999378528, y[3] -> 6.138765436688959,
y[5] -> -2.0356554447173103, y[7] -> 1.2008943866477078,
y[11] -> -4.037681072308063}},
{0.46807426662150586,
{y[2] -> 1.862568513059197, y[3] -> -1.775536571827146,
y[5] -> 1.0970815597460468, y[7] -> -3.225638584732568,
y[11] -> -0.1948912567743111}},
{0.4461504388077101,
{y[2] -> -1.8235242214443004, y[3] -> 1.703210228849542,
y[5] -> -1.0808428797366623, y[7] -> 3.5224915185675343,
y[11] -> 0.1209023444556642, y[13] -> 0.5692233220173013}},
{0.440086847216118,
{y[2] -> -8.444874723393042*10^6, y[3] -> -5.142945536843638, y[5] -> 2.6036904642523546,
y[7] -> -0.9428404374705998, y[11] -> 1.8475803844865497,
y[13] -> -0.3860579396004424}},
{0.4511481156025372,
{y[2] -> -33.76257500117095, y[3] -> 5.825293898600454,
y[5] -> -2.304204140803546, y[7] -> 0.9641080575296439,
y[11] -> -1.8210253913090075, y[13] -> 0.6120055717492042}},
{0.4879779444860755, {y[2] -> -1.9863015651557585,
y[3] -> 1.9582496199745032, y[5] -> -1.1557682101592723,
y[7] -> 2.6681947381865374, y[11] -> 0.15579441709673203,
y[13] -> 8.745313251473577}},
{0.49055456403195025,
{y[2] -> 1.8042817589822684, y[3] -> -1.9449510060407378,
y[5] -> 1.073225510039253, y[7] -> -2.8572504723288463,
y[11] -> -0.16830673515218564, y[13] -> 4.818559078817414*10^6,
y[17] -> -1.3074794690981901}},
{0.4762504985563837,
{y[2] -> -2.259128490413544, y[3] -> 2.0308673157271966,
y[5] -> -1.246733062354382, y[7] -> 2.408564076205052,
y[11] -> 0.19595756577630155, y[13] -> 7.177245398686128,
y[17] -> -0.5078441165548871}},
{0.4746150602095098,
{y[2] -> -2.174896681068332, y[3] -> 2.2450171685735034,
y[5] -> -1.1323078959490933, y[7] -> 2.768292350631055,
y[11] -> 0.34425470525998075, y[13] -> -2.226338287725374*10^7,
y[17] -> -0.19833389884098016, y[19] -> -0.9062095293690822}},
{0.46698943639051865,
{y[2] -> 2.1371340049709624,
y[3] -> -1.9384805776732492, y[5] -> 1.1707897679922044,
y[7] -> -2.5445849843960566, y[11] -> -0.1897373029487319,
y[13] -> -8.067721354487118, y[17] -> 0.277330263282675,
y[19] -> 3.324056147899286}},
{0.4797097621551765,
{y[2] -> -1.9072590570585013, y[3] -> 2.070591397093416,
y[5] -> -0.9690788472167418, y[7] -> 3.0561826931890224,
y[11] -> 0.3183539738147688, y[13] -> -1.3156973229692414*10^7,
y[17] -> -0.127013005857223, y[19] -> -4.91962778620387}},
{0.5132338128790291,
{y[2] -> -35.83900453910118,
y[3] -> 5.399617934721233, y[5] -> -2.266942040600513,
y[7] -> 0.9484611597843559, y[11] -> -1.5576669844991877,
y[13] -> 1.0764186527860076, y[17] -> -2.8298664570746155,
y[19] -> 0.26248549356388395}},
{0.5153366188102717,
{y[2] -> 18.453092415169003, y[3] -> 3.1545061540790087,
y[5] -> -3.055904152648472, y[7] -> 0.6925281833765166,
y[11] -> -1.6439498713394165, y[13] -> 0.942673947045045,
y[17] -> 2.7485136995599563*10^6, y[19] -> -0.043511090433659166,
y[23] -> -0.4227059718591429}},
{0.507702455139519,
{y[2] -> -13.882711315361206, y[3] -> 6.659369214327156,
y[5] -> -2.066791998564868, y[7] -> 1.0134881125621231,
y[11] -> -1.8355085433634732, y[13] -> 0.969857138831391,
y[17] -> -1.8677418425296055, y[19] -> 0.3524647622594828,
y[23] -> -4.976865996291514*10^6}},
{0.5096569558604533,
{y[2] -> -26.965283623861794, y[3] -> 5.5229324017247325,
y[5] -> -2.0449477704349652, y[7] -> 0.9847647557295692,
y[11] -> -1.7807824972008317, y[13] -> 0.9250175085376017,
y[17] -> -2.331175137873788, y[19] -> 0.29972607341003205,
y[23] -> -6.566392726129779}},
{0.4957655519078268,
{y[2] -> -21.27998108575053, y[3] -> 5.796751433694567,
y[5] -> -2.0849128265202017, y[7] -> 0.9881393605042436,
y[11] -> -1.823178659952556, y[13] -> 0.9312554094218648,
y[17] -> -2.180676018172768, y[19] -> 0.30854137785761665,
y[23] -> -10.457585568248941}},
{0.493353344030963,
{y[2] -> -22.153951006886118, y[3] -> 4.890785783803923,
y[5] -> -2.352952292343832, y[7] -> 0.8936217312846685,
y[11] -> -2.090595643047613, y[13] -> 0.8554583162479408,
y[17] -> -2.2984955327796883, y[19] -> 0.2731580933507656,
y[23] -> -6.311369549512267}},
{0.5395105053136733,
{y[2] -> -36.33104110633147, y[3] -> 4.757228139541344,
y[5] -> -2.443730484179421, y[7] -> 0.8507640773911131,
y[11] -> -1.991390162195122, y[13] -> 0.8673942593437886,
y[17] -> -2.662684482443947, y[19] -> 0.26566806615844113,
y[23] -> -3.043577299897147}},
{0.5263782857581697,
{y[2] -> 25.25401383144073, y[3] -> 2.923963199409909,
y[5] -> -3.767726578580592, y[7] -> 0.5176961769196098,
y[11] -> -3.9608657630391972, y[13] -> 0.5075343098739125,
y[17] -> -6.366422776649545, y[19] -> -0.05814615867207348,
y[23] -> 3756.5564429248493, y[29] -> -1.8079923822928627}},
{0.532262482824738,
{y[2] -> 8.357384007978142,
y[3] -> 2.048744093019726*10^7, y[5] -> 1.6341728378280016,
y[7] -> -1.570077546363175, y[11] -> 0.9399062116639976,
y[13] -> -1.7657426102700957, y[17] -> 1.4811650627571193,
y[19] -> -0.7524386612819524, y[23] -> 0.9429936278313875,
y[29] -> -0.06986050043371135}},
{0.5098368447944086,
{y[2] -> -4.295234386315433*10^7, y[3] -> -3.776498062431739,
y[5] -> 2.7661143772290417, y[7] -> -0.7098421573518955,
y[11] -> 2.3987498189612877, y[13] -> -0.7250012734967002,
y[17] -> 3.614761523233301, y[19] -> -0.15077284850377923,
y[23] -> 4.09568553228776, y[29] -> 0.8588293711043647,
y[31] -> -0.028361936941826257}},
{0.5079375830113259,
{y[2] -> -23.888309881712335, y[3] -> -2.7765925742896114,
y[5] -> 3.8209544810924814, y[7] -> -0.4791327626136977,
y[11] -> 4.3008436517346995, y[13] -> -0.47788409420494327,
y[17] -> 7.059371585979083, y[19] -> 0.0829753366665595,
y[23] -> -6.595269670599772*10^6, y[29] -> 1.2322052779098513,
y[31] -> 0.21494585293809093}},
{0.49555821568780584,
{y[2] -> -5.066246086172957*10^6, y[3] -> -3.630034375104794,
y[5] -> 2.8126437023535793, y[7] -> -0.6768482708550798,
y[11] -> 2.5698852965083097, y[13] -> -0.6947694877921077,
y[17] -> 3.6690972525219183, y[19] -> -0.1300624143834664,
y[23] -> 4.36421038154184, y[29] -> 0.8574541295717956,
y[31] -> 0.032379893238005227}},
{0.540002424826123,
{y[2] -> 8.362577580529036, y[3] -> 1.9433764633519504,
y[5] -> 2.4225107795045388*10^8, y[7] -> 0.04290314638959777,
y[11] -> 5.020776567428977, y[13] -> 0.019052417842858438,
y[17] -> 8.899537762149397, y[19] -> -0.5299997219297788,
y[23] -> 2.4537491123162405, y[29] -> 9.714606073017307*10^7,
y[31] -> -0.8641525290683024}},
{0.5374015273420782,
{y[2] -> -9.050943946654883, y[3] -> -2.174642714111875,
y[5] -> -64.72627860205496, y[7] -> -0.04107479112407146,
y[11] -> -5.64346746363049, y[13] -> -0.10235554558065739,
y[17] -> -5.558626478659901*10^6, y[19] -> 0.4653174372715049,
y[23] -> -2.4615182593159406, y[29] -> -137.80046944118646,
y[31] -> 0.7701814129443211}},
{0.5538027559050895,
{y[2] -> -6.785104626007184, y[3] -> -1.9959279874923013,
y[5] -> -2.7752793143328356, y[7] -> 0.3104132251148657,
y[11] -> -1.5389202440305818, y[13] -> 0.2609153564948118,
y[17] -> -5.665173002982876, y[19] -> 0.842488062643456,
y[23] -> -0.9547285835778841, y[29] -> -3.486241866347132,
y[31] -> 0.8694448889517893}},
{0.5410949110097839,
{y[2] -> -3.4651348668987656, y[3] -> -4.517992436770534,
y[5] -> -0.9598436878013499, y[7] -> 1.3900674219713567,
y[11] -> -0.7452554784599054, y[13] -> 5.922283371582142,
y[17] -> 0.09665046348278163, y[19] -> 3.345899657456896,
y[23] -> -0.27063629346385637, y[29] -> 0.38458490068651696,
y[31] -> -4.992562635912787*10^6, y[37] -> -0.7571633316839521}},
{0.5429973547947488,
{y[2] -> -6.844959654454303,
y[3] -> -2.0091872017143815, y[5] -> -3.4801189326290003,
y[7] -> 0.22523188359846552, y[11] -> -1.8644151887734883,
y[13] -> 0.1895280833052751, y[17] -> -7.30373294446857,
y[19] -> 0.7032766359810382, y[23] -> -1.1816031941841862,
y[29] -> -3.735753757203667, y[31] -> 0.9750780803884066,
y[37] -> -0.002931194578079365}},
{0.5467999587431224,
{y[2] -> 1.2962215491488913, y[3] -> -3.289397771850493,
y[5] -> 0.2771825974157378, y[7] -> -14.718144387520871,
y[11] -> -1.137514653828102, y[13] -> 1.6894017049514538,
y[17] -> -0.8181503184032657, y[19] -> 0.6295688520934862,
y[23] -> -2.6466666223193154, y[29] -> 3.089692195396208,
y[31] -> -0.5608569643685216, y[37] -> 0.49683613415562555}},
{0.5559476548255569,
{y[2] -> -9.848512412603167,
y[3] -> -2.50605833982349, y[5] -> -6.125047625698561,
y[7] -> 0.060113588218360284, y[11] -> -3.431084432122056,
y[13] -> -0.03795275182176026, y[17] -> -2.7094695327229677*10^6,
y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823,
y[29] -> 1.2680941513401158*10^7, y[31] -> 0.5666075600978651,
y[37] -> -0.0937132554032237}},
{0.5505709016488777,
{y[2] -> 1.9978234885858357, y[3] -> -2.6734434483822795,
y[5] -> 0.999575779341089, y[7] -> -1.4977353323799243,
y[11] -> -3.47473672784074*10^6, y[13] -> -0.6567175708714164,
y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*10^6,
y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913,
y[31] -> -1.9302319705248326*10^6, y[37] -> 4.090882786292679,
y[41] -> -0.5196037223519064}},
{0.5410183504459096,
{y[2] -> 6.812083492495216, y[3] -> 2.0560405801391326,
y[5] -> 3.2692616497530045, y[7] -> -0.258232265287513,
y[11] -> 1.85743590978529, y[13] -> -0.15763741874306123,
y[17] -> 8.362997598365784, y[19] -> -0.7854226727043914,
y[23] -> 1.0612568625374965, y[29] -> 3.420494830428003,
y[31] -> -0.9285855774319701, y[37] -> -0.3976871295336513,
y[41] -> 0.8078091963254634}},
{0.5419769095397796,
{y[2] -> 3.556992190874782, y[3] -> 4.224165246660858,
y[5] -> 1.0011346175495501, y[7] -> -1.362608171502796,
y[11] -> 0.7198577138179995, y[13] -> -8.78355646306516,
y[17] -> -0.11487864115928048, y[19] -> -3.076071739644593,
y[23] -> 0.32659199253704546, y[29] -> -0.38089384311933716,
y[31] -> 5.116561470832145*10^6, y[37] -> 3.090373695813483,
y[41] -> 0.4610508675579661, y[43] -> -0.316274724263519}},
{0.5447128396652148,
{y[2] -> 3.5104534047833686,
y[3] -> 4.340718160982424, y[5] -> 0.9818870515462708,
y[7] -> -1.317408707534025, y[11] -> 0.729445988142171,
y[13] -> -6.911342051144954, y[17] -> -0.10690390484592481,
y[19] -> -2.8606929480231145, y[23] -> 0.33402599154861323,
y[29] -> -0.3873919646374986, y[31] -> 9.948718836620959*10^6,
y[37] -> 2.6736163206571177, y[41] -> -0.02188732982516384,
y[43] -> 4.143328442778921}},
{0.5621737511465918,
{y[2] -> 3.0969408474533338, y[3] -> 3.373238071074912,
y[5] -> 0.7943086850031457, y[7] -> -1.7100968766160136,
y[11] -> 0.46969552238862494, y[13] -> -17.92344438105851,
y[17] -> -0.3190567780751474, y[19] -> 4586.317756875522,
y[23] -> -0.012841815686448831, y[29] -> -0.640979574139176,
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