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

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. is a real primitive character modulo D is an element of N with L(1, psi) << (log D)(-25-epsilon), then, for any prime q is an element of [D-300, D-O(1)], one has L(1/2, (chi)) not equal 0 for almost all Dirichlet characters chi (mod q).

Automated summary

We study the non-vanishing property of Dirichlet L-functions at the central point, assuming the existence of an exceptional Dirichlet character. In particular, we prove that if ψ is a real primitive character modulo D, with L(1, ψ) << (log D)(-25-ε), then for any prime q in [D-300, D-O(1)], L(1/2, χ) is not equal to 0 for almost all Dirichlet characters χ (mod q).