Explicit zero-free regions for Dirichlet L-functions

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Abstract

Let Lk(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 Lk(S). The first result is that Lk(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 χ1 and χ2 are distinct real primitive characters modulo k1 and k2, respectively, and if β1 is a zero of L(s, χi), i = 1, 2, then min 1, β2} < 1 − 1(R1 log M1), where R1 = (5 − √5)(15 − 10√2), and M1 = max{k1k217, 13}.

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