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]</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-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 \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]

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]

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,...,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]}}

{ {0.5, {y[2] -> -1.462885206912493*^8}}, {0.4999999999999996, {y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}}, {0.4999999999999998, {y[2] -> -3.910648170382393*^8, y[3] -> 7.310011357069776*^8}}, {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*^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*^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*^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*^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*^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*^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*^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*^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*^6, y[29] -> 1.2322052779098513, y[31] -> 0.21494585293809093}}, {0.49555821568780584, {y[2] -> -5.066246086172957*^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*^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*^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*^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*^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*^6, y[19] -> 0.4894609605457892, y[23] -> -1.8252237433726823, y[29] -> 1.2680941513401158*^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*^6, y[13] -> -0.6567175708714164, y[17] -> 0.2855245128964309, y[19] -> -3.581256310702213*^6, y[23] -> 0.5126720691895783, y[29] -> -0.5586213537787913, y[31] -> -1.9302319705248326*^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*^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] -> 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3.489295385976381, y[61] -> -1.205411995765, y[67] -> -0.40674063282043443, y[71] -> 153310.1169285756, y[73] -> 0.2220413097490152}}, {0.5386875035297409, {y[2] -> 1949.9397534072007, y[3] -> -3.1383625037232807, y[5] -> 15.34067180380558, y[7] -> -0.1076496770192988, y[11] -> 55167.33853812988, y[13] -> -0.37367290049542834, y[17] -> 4.837463608290301, y[19] -> 0.26746159448234225, y[23] -> 7008.4674573165275, y[29] -> -31284.91278677744, y[31] -> 0.20750551257516886, y[37] -> -0.4867986451359988, y[41] -> 614673.136216316, y[43] -> 0.2254964384983165, y[47] -> 1.2982364795443602, y[53] -> -33590.8429222153, y[59] -> 3.9903939619803683, y[61] -> -1.0843000310863584, y[67] -> -0.40211189296375205, y[71] -> 4.405057527315247, y[73] -> 0.0007043557151270179}}, {0.5579498988797087, {y[2] -> 2.2382040199746585, y[3] -> 2.5912101493453545, y[5] -> 0.5481313647466814, y[7] -> -3.224577081186167, y[11] -> 0.15728385738525189, y[13] -> 4.123851838318665, y[17] -> -0.5329060728267422, y[19] -> 10.19583186955811, y[23] -> -0.3969485275960606, y[29] -> -0.8250765889045757, y[31] -> 2.22223392731537, y[37] -> 0.6792223938404973, y[41] -> -0.646843188596205, y[43] -> 3.0457854238243334, y[47] -> 8192.345063681541, y[53] -> -1.0434194839068849, y[59] -> -3.170896012271207, y[61] -> -0.16833878264728688, y[67] -> 0.16405596604881661, y[71] -> -2.9086942449735034, y[73] -> 0.5613454008457452}}, {0.5391983432183001, {y[2] -> 238.0940917012289, y[3] -> -3.1586997158152923, y[5] -> 46.275900458252416, y[7] -> -0.09210237318381041, y[11] -> 1839.4173845777152, y[13] -> -0.34705809680847627, y[17] -> 5.444413456786553, y[19] -> 0.22071592194342135, y[23] -> 141.17003292303258, y[29] -> -1072.8082559288473, y[31] -> 0.22874240679862948, y[37] -> -0.5204089962232741, y[41] -> 906.0420776048169, y[43] -> 0.2198389174538509, y[47] -> 1.299492534698202, y[53] -> -40.8775854214683, y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588, y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234, y[73] -> -0.08134651616351249}} }