Riemann Hypothesis and Random Walks: the Zeta case

TL;DR

This paper explores the connection between the Riemann Hypothesis, random walk behavior of series related to primes, and the argument of the zeta function, proposing new methods for zero computation and probabilistic modeling.

Contribution

It extends previous work linking prime-based series to the Riemann Hypothesis to the zeta function and introduces a novel algorithm for high-precision zero calculation.

Findings

Validated a new algorithm computing the 10^100 zero to over 100 digits

Developed a probabilistic model for Riemann zeros based on correlation functions

Extended the random walk approach to the Riemann zeta function

Abstract

In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the critical line $\Re (s) > \tfrac{1}{2}$, and the Riemann Hypothesis for this class of $L$-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet $L$-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a 1-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely $10^{100}$-th zero to over 100 digits, far beyond what is currently known.