I am trying to plot this
which is a numerical simulation of the Montgomery-Odlyzko law for the nontrivial 1st $10^5$ zeros of the Riemann zeta function $ζ(s)$. The solid line is given by 1-(Sin[π x]/(πx))^2
.
The following is what I can't seem to interpret:
The blue symbols indicate the pair correlation function of normalized spacings $δn=(γ_{n+1}-γ_n)\log(γ_n/2π)/2π$ between two consecutive nontrivial zeros $1/2+iγ_n$ and $1/2+iγ_{n+1},\ n=1\dots10^5)$ of the Riemann zeta function $ζ(s)$.
from this page.
I have looked at various ways of calculating the density, using Histogram
, NearestFunction
, FindClusters
, etc. but am getting nothing like the image above. I realise that this is largely due to a conceptual misunderstanding, but I was hoping someone could point me in the right direction.
Update
plotted with
DeleteCases[Flatten[Table[Abs[z1 - z2] Log[z2/(2 π)]/(2 π),
{z1, Take[zz,1000]}, {z2, Take[zz,1000]}]], 0];
Show[ListPlot[HistogramList[%, {0, 3, 0.1}][[2]]/200,
DataRange -> 3], Plot[1 - (Sin[π u]/(π u))^2, {u, 0, 3}]]
using First $10^5$ zeros as zz
, thanks to Rahul Narain's comments below. Unfortunately, my computer won't calculate for $10^5$ zeros, but link left for those that will.
... I am now beginning to appreciate the computational power that would have to be used to generate an image like this or this!
zzd = Flatten@Table[Abs[zz[[i]] - zz[[j]]], {i, 2, 1000}, {j, 1, i - 1}]; Histogram[zzd, {0, 3, 0.2}]
i.stack.imgur.com/Pc3Tw.png $\endgroup$ – user484 Jun 5 '14 at 19:26