TITLE: Zeta zeros standard normal distribution about $\vartheta (\gamma_n)$
QUESTION [6 upvotes]: Asked at MSE here without response.
I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon.
That said, below is a partially scaled histogam plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for the first $2$m zeros:

It certainly looks to have standard normal distribution, is this indeed the case? Does it depend on the RH, or is it independent of it?

REPLY [8 votes]: What you're observing is a remarkable theorem of Selberg.  The usual notation is to let $N(T)$ denote the number of zeros of $\zeta(s)$ with ordinates between $0$ and $T$.  Then the argument principle (with Stirling's formula) gives 
$$
N(T) = \frac{T}{2\pi} \log \frac{T}{2\pi e} + \frac{7}{8} + S(T) + O\Big(\frac 1T\Big),  
$$ 
where $S(T) = \frac {1}{\pi} \text{arg} \zeta(\tfrac 12+iT)$ (and the argument is 
defined by continuous variation along line segments from $2$ to $2+iT$ and then from $2+iT$ to $1/2+iT$).  Then Selberg showed that $\pi S(T)=\text{arg}\zeta(\tfrac 12+iT)$ has a normal distribution with mean $\sim 0$ and variance $\sim \frac{1}2 \log \log T$.  What you're looking at is the distribution of this remainder term in the asymptotic for the number of zeros when evaluated at the ordinates of the $n$-th zero, and from Selberg's work one can deduce that this is Gaussian.  You'll find a discussion of Selberg's theorem in the books of Edwards or Titchmarsh.