To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand

to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process?

Yet, what I figured out from readnigs, is that such a process acts only (conjectured to be) at the level of the neighborhood of the zeros. Not sure how neighborhood can be interpreted, or is the radius even larger?

• What do you mean by 'to which extent"? – Kunnysan Jul 16 '13 at 17:17
• I mean the distance or radius (or some metric) from a non-trivial zero on the $1/2$ axis. There is no philosophy behind enxtent. – al-Hwarizmi Jul 16 '13 at 17:25

Assume the Riemann Hypothesis. Let $\rho = \tfrac 12 + i\gamma$ denote zeros and $N(T)$ be the number of zeros with $\gamma \in [T,2T]$. Let $\psi = \psi(T)$ be a function tending to infinity with $t$, arbitrarily slowly. One can prove that, $$\frac{1}{N(T)}\# \bigg \{ T \leq \gamma \leq 2T: \frac{N \big (\gamma + \frac{\pi \psi}{\log T} \big )- N \big (\gamma - \frac{\pi \psi}{\log T} \big ) - \psi}{\sqrt{\psi}} \in [\alpha,\beta] \bigg \} \rightarrow \int_{\alpha}^{\beta} e^{-u^2/2} \frac{du}{\sqrt{2\pi}}$$ Maybe I got the constant in the variance slightly wrong. The statement is no longer true if $\psi$ does not tend to infinity.

As a start I recommend looking at http://arxiv.org/abs/1205.0303 and http://arxiv.org/abs/math.nt/0208220. To prove the statement I wrote above take a look at Fujii's work cited as Theorem 1 in Brad Rodgers paper and modify Fujii's proof by using Gonek's lemma to sum over the zeros. A version of Gonek's Lemma is stated in equation (1.6) of http://arxiv.org/abs/0805.2745