A note on prime number races and zero free regions for $L$ functions

TL;DR

This paper establishes a connection between the sign changes of certain prime sum functions associated with Dirichlet characters and the zero-free regions of their corresponding $L$-functions, contributing to understanding prime number races.

Contribution

It proves that finite sign changes in prime sum partial sums imply the existence of a zero-free region for the associated $L$-function.

Findings

Finite sign changes imply zero-free regions for $L$-functions.

Provides criteria linking prime sum behavior to zeros of $L$-functions.

Advances understanding of prime number races and zero distribution.

Abstract

Let $\chi$ be a real and non-principal Dirichlet character, $L(s,\chi)$ its Dirichlet $L$-function and let $p$ be a generic prime number. We prove the following result: If for some $0\leq \sigma<1$ the partial sums $\sum_{p\leq x}\chi(p)p^{-\sigma}$ change sign only for a finite number of $x$, then there exists $\epsilon>0$ such that $L(s,\chi)$ has no zeros in the half plane $Re(s)>1-\epsilon$.