Generalized Riemann Hypothesis and Stochastic Time Series

TL;DR

This paper extends the domain of convergence for Dirichlet $L$-functions to the critical line using probabilistic and ergodic methods, suggesting a stochastic interpretation of prime residue distributions.

Contribution

It introduces a novel approach linking stochastic time series analysis and ergodic theory to the convergence of Dirichlet $L$-functions at the critical line.

Findings

Convergence domain extends to Re(s) > 1/2 without zeros.

Dirichlet character sums behave like diffusive random walks.

Ergodic and stationary properties are established via ensemble averaging.

Abstract

Using the Dirichlet theorem on the equidistribution of residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes, we show that the domain of convergence of the infinite product of Dirichlet $L$-functions of non-principal characters can be extended from $\Re(s) > 1$ down to $\Re(s) > \half$, without encountering any zeros before reaching this critical line. The possibility of doing so can be traced back to a universal diffusive random walk behavior $C_N = {\cal O}(N^{1/2})$ of the series $C_N = \sum_{n=1}^N \chi (p_n)$ over the primes $p_n$ where $\chi$ is a Dirichlet character, which underlies the convergence of the infinite product of the Dirichlet functions. The series $C_N$ presents several aspects in common with stochastic time series and its control requires to address a problem similar to the Single Brownian Trajectory Problem in statistical mechanics. In the case of the Dirichlet functions of non principal characters, we show that this problem can be solved in terms of a self-averaging procedure based on an ensemble $\CE$ of block variables computed on extended intervals of primes. Those intervals, called {\em inertial intervals}, ensure the ergodicity and stationarity of the time series underlying the quantity $C_N$. The infinity of primes also ensures the absence of rare events which would have been responsible for a different scaling behavior than the universal law $C_N = {\cal O}(N^{1/2})$ of the random walks.