Exceptional characters and nonvanishing of Dirichlet L-functions

Let. be a real primitive character modulo D. If the L-function L(s, psi) has a real zero close to s = 1, known as a Landau-Siegel zero, then we say the character psi is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values L(1/2, chi) of the Dirichlet L-functions L(s, chi) are nonzero, where. 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., the function L(s, chi) has at most a simple zero at s = 1/2.

Automated summary

Assuming the existence of exceptional characters, we have proven that at least fifty percent of the central values of the Dirichlet L-functions are nonzero, and for most cases, the function has at most a simple zero at s=1/2.