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?

Can someone help me to figure out what is proven or conjectured about this matter.

  • $\begingroup$ What do you mean by 'to which extent"? $\endgroup$ – Kunnysan Jul 16 '13 at 17:17
  • $\begingroup$ 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. $\endgroup$ – 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


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