Statistic of zeros of Riemann auxiliary function

TL;DR

This paper presents a statistical analysis of the zeros of the Riemann auxiliary function, based on extensive computational data, revealing that approximately 63.9% of these zeros lie to the left of the critical line.

Contribution

The paper provides the first large-scale statistical analysis of zeros of the Riemann auxiliary function using over 162,000 computed zeros.

Findings

63.9% of zeros satisfy β<1/2

All zeros computed with 25 decimal digits accuracy

Zeros distributed with notable bias towards the left of the critical line

Abstract

We have computed all zeros $\beta+i\gamma$ of $\mathop{\mathcal R }(s)$ with $0<\gamma<215946.3$. A total of 162215 zeros with 25 correct decimal digits. In this paper we offer some statistic based on this set of zeros. Perhaps the main interesting result is that $63.9\%$ of these zeros satisfies $\beta<1/2$.