Omega(N)

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]\displaystyle{ 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) = \inf_{y_i\in {\mathbb R}} \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] at [math]\displaystyle{ y_1=0, y_2=\infty, y_3=\infty }[/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, 
      y[31] -> 7.6282899845236045, y[37] -> 1.4992499178271976, 
      y[41] -> 0.07388970007557207, y[43] -> -1077.9135012938261}}, 

{0.5458049553223693,

      {y[2] -> -7.626649842911723, 
      y[3] -> 3.182773873173586*10^7, y[5] -> -1.9670364421721978, 
      y[7] -> 0.7352144275407777, y[11] -> -1.8239756295292955, 
      y[13] -> 1.772186178431415, y[17] -> -0.5662015719080454, 
      y[19] -> 0.698021316660052, y[23] -> -1.7240664629104805, 
      y[29] -> -0.5072752073185517, y[31] -> 0.9583259450451189, 
      y[37] -> 4.262348851794279, y[41] -> -1.5807084927110222, 
      y[43] -> 0.5802847144626384}}, 

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}