Generalized Riemann Hypothesis, Time Series and Normal Distributions

TL;DR

This paper investigates the analytic continuation of Dirichlet L-functions related to the Generalized Riemann Hypothesis by analyzing the stochastic properties of series involving primes and Dirichlet characters, revealing diffusive behavior that supports extending their domain of convergence.

Contribution

It demonstrates that the series associated with non-principal Dirichlet characters exhibit diffusive random walk behavior, enabling extension of the L-functions' domain of convergence to the critical line.

Findings

Series $B_N$ follow normal distribution laws.

Series $B_N$ behave as diffusive random walks $O( oot{2}{N})$.

Domain of convergence extends to $ ext{Re}(s)=1/2$ without zeros.

Abstract

$L$ functions based on Dirichlet characters are natural generalizations of the Riemann $\zeta(s)$ function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. In this paper we address the Generalized Riemann Hypothesis relative to the non-trivial complex zeros of the Dirichlet $L$ functions by studying the possibility to enlarge the original domain of convergence of their Euler product. The feasibility of this analytic continuation is ruled by the asymptotic behavior in $N$ of the series $B_N = \sum_{n=1}^N \cos \bigl( t \log p_n - \arg \chi (p_n) \bigr)$ involving Dirichlet characters $\chi$ modulo $q$ on primes $p_n$. Although deterministic, these series have pronounced stochastic features which make them analogous to random time series. We show that the $B_N$'s satisfy various normal law probability distributions. The study of their large asymptotic behavior poses an interesting problem of statistical physics equivalent to the Single Brownian Trajectory Problem, here addressed by defining an appropriate ensemble $\mathcal{E}$ involving intervals of primes. For non-principal characters, we show that the series $B_N$ present a universal diffusive random walk behavior $B_N = O(\sqrt{N})$ in view of the Dirichlet theorem on the equidistribution of reduced residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes. This purely diffusive behavior of $B_N$ implies that the domain of convergence of the infinite product representation of the Dirichlet $L$-functions for non-principal characters can be extended from $\Re(s) > 1$ down to $\Re (s) = \frac{1}{2}$, without encountering any zeros before reaching this critical line.