A note on exceptional characters and non-vanishing of Dirichlet L-functions

Abstract

<p>We study non-vanishing of Dirichlet <em>L</em>-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if ψ is a real primitive character modulo <em>D</em> ∈ N with <em>L</em>(1,ψ) ≪ (log<em> D</em>)−^25−ε, then, for any prime <em>q </em>∈ [<em>D</em>³⁰⁰, <em>D</em>^O(1)], one has <em>L</em>(1/2, χ) ≠ 0 for almost all Dirichlet characters χ (mod <em>q</em>).<br></p>