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 $0\le \sigma <1$ the partial sums ${\sum }_{p\le x}\chi (p)p^{-\sigma }$ 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 $Re(s)>1-\epsilon$.

9 pages, revised version