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

Abstract

We study non-vanishing of Dirichlet L-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if $$\psi $$ ψ is a real primitive character modulo $$D \in \mathbb {N}$$ D ∈ N with $$L(1, \psi ) \ll (\log D)^{-25-\varepsilon }$$ L ( 1 , ψ ) ≪ ( log D ) - 25 - ε , then, for any prime $$q \in [D^{300}, D^{O(1)}]$$ q ∈ [ D 300 , D O ( 1 ) ] , one has $$L(1/2, \chi ) \ne 0$$ L ( 1 / 2 , χ ) ≠ 0 for almost all Dirichlet characters $$\chi \, (\textrm{mod} \, q)$$ χ ( mod q ) .