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

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 $Re(s)>1-\epsilon$.