Exceptional characters and nonvanishing of Dirichlet $L$-functions

Abstract

Let ψ be a real primitive character modulo D. If the L-function L(s, ψ) has a real zero close to s = 1, known as a Landau-Siegel zero, then we say the character ψ is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values L(1/2, χ) of the Dirichlet L-functions L(s, χ) are nonzero, where χ ranges over primitive characters modulo q and q is a large prime of size D O(1) . Under the same hypothesis we also show that, for almost all χ, the function L(s, χ) has at most a simple zero at s = 1/2.