On the statistics of differences of zeta zeros starting from zero number $10^{23}$

TL;DR

This paper analyzes the statistical properties of differences between high-lying Riemann zeta zeros, revealing consistent patterns in their distributions, skewness, and variance, even at zeros as high as 10^{23}.

Contribution

It provides the first detailed statistical analysis of zeta zero differences at extremely high zeros, uncovering patterns in skewness, variance, and distribution shape.

Findings

Distributions of zero differences are generally skewed towards the nearest zero.

Variance of differences shows local maxima or minima at each zero.

Distributions can be modeled with Johnson probability density functions.

Abstract

We study distributions of differences of unscaled Riemann zeta zeros, $\gamma-\gamma^{'}$, at large. We show, that independently of the location of the zeros, i.e., even for zeros as high as $10^{23}$, their differences have similar statistical properties. The distributions of differences are skewed usually towards the nearest zeta zero. We show, however, that this is not always the case, but depends upon the distance and number of nearby zeros on each side of the corresponding distribution. The skewness, however, always decreases when zeta zero is crossed from left to right, i.e., in increasing direction. Furthermore, we show that the variance of distributions has local maximum or, at least, a turning point at every zeta zero, i.e., local minimum of the second derivative of the variance. In addition, it seems that the higher the zeros the more compactly the distributions of the differences are located in the skewness-kurtosis -plane. Furthermore, we show that distributions can be fitted with Johnson probability density function, despite the value of skewness or kurtosis of the distribution.