Explicit zero-free regions for Dirichlet L-functions

Abstract

Let L k ( S) be the product of the ϕ( k) Dirichlet L-functions formed with characters modulo k. We prove the existence of explicit numerical zero-free regions for L k ( S). The first result is that L k ( S) has at most a single zero in the region { s: σ > 1 − 1 (R log M) } , where R = 9.645908801 and M = max { k, k | t|, 10}. The only possible zero in this region is a simple real zero arising from an L-function formed with a real non-principal character. The second result is that if χ 1 and χ 2 are distinct real primitive characters modulo k 1 and k 2, respectively, and if β 1 is a zero of L( s, χ i ), i = 1, 2, then min {β 1, β 2} < 1 − 1 (R 1 log M 1) , where R 1 = (5 − √5) (15 − 10√2) , and M 1 = max{ k 1k 2 17 , 13} .