, Volume 157, Issue 3-4, pp 575-604

On fluctuations of Riemann’s zeta zeros

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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 $x\sim \left( \log k\right) ^{\beta }, \beta >0,$ for large $k$ follow the two-variate Gaussian distribution with correlation $\left( 1-\beta \right) _{+}\! .$