Wolfram Language

Matrices aléatoires

Zéros de la fonction zeta de Riemann

Les zéros de la fonction zêta de Riemann ont été conjecturé pour être liés aux valeurs propres des opérateurs et matrices hermitiens. Comparez la distance normalisée des zéros à la distance normalisée des valeurs propres d'échantillonnage de GaussianUnitaryMatrixDistribution.

In[1]:=
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dim = 100;

Définissez la distribution de la propriété de la matrice pour la moitié des valeurs propres ordonnées.

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

Générez un échantillon aléatoire de la distribution.

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

Calculez la densité de séparation en utilisant une distribution non paramétrique.

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

Les zéros de la fonction zêta dans la ligne critique à partir de la th zero (Odzlyko).

In[6]:=
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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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Comparez l'histogramme de densité de la densité normalisée des zéros à la densité du noyau de la distribution lissée de kernel.

Montrer l'entrée complète de 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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