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]:=
dim = 100;
Defina a distribuição da propriedade de matriz para a metade central dos valores próprios ordenados.
In[2]:=
\[ScriptCapitalD] =
MatrixPropertyDistribution[
Take[Sort[Eigenvalues[x]], {dim/4, 3/4*dim}],
x \[Distributed] GaussianUnitaryMatrixDistribution[dim]];
Gere uma amostra aleatória da distribuição.
In[3]:=
eigs = RandomVariate[\[ScriptCapitalD], 10^3];
Calcule a densidade de separação usando uma distribuição não paramétrica.
In[4]:=
fun[ev_] := Differences[ev] Sqrt[4 dim - Most[ev]^2]
In[5]:=
spacingdensity =
SmoothKernelDistribution[1/(2 Pi) Flatten[fun /@ eigs]]
Out[5]=
Os zeros da função zetana linha crítica começando no th zero (Odzlyko).
In[6]:=
zeros = TemporalData[EventSeries, {CompressedData["
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"], {{0, 9999, 1}}, 1, {"Discrete", 1}, {
"Discrete", 1}, 1, {ResamplingMethod -> None}}, False, 10.3];
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
Out[7]=