On Fluctuations of Riemann's Zeta Zeros

Abstract

It is shown that the normalized fluctuations of Riemann's zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, $\gamma _{k}$ and $\gamma _{k+x},$ with $x\sim(\log k)^{\beta}$, $\beta >0$, for large $k$ follow the two-variate Gaussian distribution with correlation $(1-\beta)_{+}$.