Notes on the Phase Statistics of the Riemann Zeros

TL;DR

This paper numerically analyzes the phase statistics of the Riemann zeta function's zeros, focusing on the imaginary part of the logarithmic derivative, to understand their distribution and behavior.

Contribution

It introduces a novel numerical method for studying the phase statistics of Riemann zeros by analyzing the logarithmic derivative along specific paths.

Findings

Identifies patterns in the phase distribution of the zeta function zeros.

Provides numerical evidence supporting conjectures about zero distribution.

Enhances understanding of the zeta function's behavior near zeros.

Abstract

We numerically investigate, for zeros $\rho=1/2+i\gamma$, the statistics of the imaginary part of $\log(\zeta^\prime(1/2+i\gamma))$, computed by continuous variation along a vertical line from $\sigma=4$ to $4+i\gamma$ and then along a horizontal line to $1/2+i\gamma$.