# Wolfram Language™

## Zeros of the Riemann Zeta Function

The zeros of the Riemann zeta function have been conjectured to be related to the eigenvalues of Hermitian operators and matrices. Compare the normalized spacing of the zeros to the normalized spacing of the bulk eigenvalues of samples from GaussianUnitaryMatrixDistribution.

In[1]:=
`dim = 100;`

Define matrix property distribution for the central half of the sorted eigenvalues.

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

Generate a random sample from the distribution.

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

Compute spacing density using nonparametric distribution.

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

The zeros of the zeta function in the critical line starting at the th zero (Odzlyko).

In[6]:=
```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 the density histogram of the normalized spacing of the zeros to the estimated spacing density of the smooth kernel distribution.

show complete Wolfram Language input
In[7]:=
```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]=