An explicit zero-free region for the Dirichlet L-functions

Abstract

Let $L(s,\chi )$ be the Dirichlet $L$-function associated to a non-principal primitive character $\chi$ modulo $q$ with $3\le q\le 400000$. We prove a new explicit zero-free region for $L(s,\chi )$: $L(s,\chi )$ does not vanish in the region $\Re s\ge 1-\frac{1}{R\log \left(q\max (1,|\Im s|)\right)}$ with $R=5.60$. This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$.

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• Explicit formulae for primes in arithmetic progressions, II