Prime zeta function statistics and Riemann zero-difference repulsion

TL;DR

This paper derives a statistical explanation for the observed repulsion between the differences of Riemann zeros and the zeros themselves, based on prime zeta function properties and their covariance structure.

Contribution

It introduces a novel derivation linking prime zeta function statistics to zero-difference repulsion in the Riemann zeta function.

Findings

Prime zeta function on the critical line is asymptotically normally distributed.

Negative covariance occurs at separations near Riemann zero imaginary parts.

The method generalizes to zeros of related L-functions.

Abstract

We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical "repulsion" between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function's magnitude on the 1-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related L-functions.