On Fluctuations of Riemann's Zeta Zeros

TL;DR

This paper demonstrates that the normalized fluctuations of Riemann zeta zeros follow a Gaussian distribution and explores their joint fluctuations, revealing a correlation structure depending on the spacing parameter.

Contribution

It establishes the Gaussian nature of zero fluctuations and characterizes their joint distribution for zeros separated by specific logarithmic scales.

Findings

Zeros' fluctuations are Gaussian distributed.

Joint fluctuations of zeros follow a bivariate Gaussian distribution.

Correlation depends on the logarithmic scale parameter.

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 $(1-\beta)_{+}$.