Exceptional characters and nonvanishing of Dirichlet L-functions

2020/12/08 by Hung M. Bui, Kyle Pratt, Bui, H. M. +3
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.04392

openalex publication_date 2020/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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