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

Abstract

Let [Formula: see text] be a real and non-principal Dirichlet character, [Formula: see text] its Dirichlet [Formula: see text]-function and let [Formula: see text] be a generic prime number. We prove the following result: If for some [Formula: see text] the partial sums [Formula: see text] change sign only for a finite number of integers [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] has no zeros in the half plane [Formula: see text].