Wolfram Language

Matrizes aleatórias

Zeros da função zeta de Riemann

É conjeturado que os zeros da função zeta de Riemann estão relacionados com os valores próprios de operadores e matrizes hermitianas. Compare a separação normalizada dos zeros com a separação normalizada dos valores próprios em conjunto das amostras de GaussianUnitaryMatrixDistribution.

In[1]:=
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dim = 100;

Defina a distribuição da propriedade de matriz para a metade central dos valores próprios ordenados.

In[2]:=
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\[ScriptCapitalD] = MatrixPropertyDistribution[ Take[Sort[Eigenvalues[x]], {dim/4, 3/4*dim}], x \[Distributed] GaussianUnitaryMatrixDistribution[dim]];

Gere uma amostra aleatória da distribuição.

In[3]:=
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eigs = RandomVariate[\[ScriptCapitalD], 10^3];

Calcule a densidade de separação usando uma distribuição não paramétrica.

In[4]:=
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fun[ev_] := Differences[ev] Sqrt[4 dim - Most[ev]^2]
In[5]:=
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spacingdensity = SmoothKernelDistribution[1/(2 Pi) Flatten[fun /@ eigs]]
Out[5]=

Os zeros da função zetana linha crítica começando no th zero (Odzlyko).

In[6]:=
Click for copyable input
zeros = TemporalData[EventSeries, {CompressedData[" 1:eJws03c8l9//x3GjUlFa6FOSaBhR0ZCGhkq0qVARhQipNEQq0hJlpSilQalQ oakQ0VCpaBgtlEilEiLf3+88/OF2v13nOq73Oa/X8zXAfs0CBykJCQnJ//vr NkRCwsd1ufUljblT7BNx0ktcVIu6veyEh8bgElN86oI+uzHlIr543eZPrG9n L7w3GKUc0MsbvxzCIccwMBPHVuPyjitYV8fUubjPG/Wj8XQCxr/C1gYslV0p rBmFMmZouRXrj2DSbdQtQYV/OGaQg7BwNKZboZo/rjyB5+6ifznqdnIUru6P WdPwuxtWHMYD11GhGq3q8dQgJ+EEHfSYjmo2eDYYD17AywWo+QUvSa4Shk7E sHm4cA2G78TPyWj8FI1qcYKCs9BZD88uwAo39NuH766hZj1+7uIivDsSN9hi VzcMPIg6caiaj0/L8F4TfjJcLRxlhjftce82vBqLTzOx7zfU6+0q3DYMmy1x jSPWBOLdW/ikFEfUYIWiG+fSwxOWWLce7+zCk/H4Ng8lXmNLH3fqZYqLluFM T+x8uu39YzR+j/od1wgbNHHkHBzkhbP34580rLqN6yqxpreHMK8/as1HVRcc GIVqV7DkDXaub9vfbq2wU0+8qI8Ntqi0DR3CcPBr3NNuHXXui7cmo9oSnOmM 36NROg09SnBAp/XCZ8oYMQV3rMSl2/FsNMrdx/kNeLiDp3CnMh4Yh28c8VcI 7kvAyHuYXI9+chvYr45HjfHiRgyMxA7JuPg2mv7A2N4b6fMI/GeCETZofBBn nEGLW23rPzBPYRN9Ho3rpmGHzegShTfvoOEv7C25WXhkNMYtwpTNuPcgjriI 8SmY9hMXdPIS9lRGm5k4xhqPrsePx/FNBo74gqbyW4QvhmH1YmxZgT124Zbz 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Compare o histograma de densidade da separação normalizada de zeros com a densidade de separação estimads da distribuição do kernel suave.

mostre o input completo da Wolfram Language
In[7]:=
Click for copyable input
zlist = zeros["Values"]; histogram = Histogram[Differences[zlist]/(2 Pi) Log[Most[zlist]/(2 Pi)], 50, PDF, PlotTheme -> "Detailed"]; plot = Plot[PDF[spacingdensity][x], {x, 0, 2.5}]; Show[Legended[histogram, SwatchLegend[{ColorData[97, 2]}, {"zeros"}]], Legended[plot, SwatchLegend[{ColorData[97, 1]}, {"spacing density"}]]]
Out[7]=

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