TY - JOUR
T1 - Exceptional characters and nonvanishing of Dirichlet L-functions
AU - Bui, Hung
AU - Pratt, Kyle
AU - Zaharescu, Alexandru
N1 - Funding Information:
During part of this work the second author was supported by the National Science Foundation Graduate Research Program under grant number DGE-1144245. The authors thank the anonymous referee for their comments on a previous version of this manuscript.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2021/6
Y1 - 2021/6
N2 - 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 DO(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.
AB - 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 DO(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.
U2 - https://doi.org/10.1007/s00208-020-02136-9
DO - https://doi.org/10.1007/s00208-020-02136-9
M3 - Article
VL - 380
SP - 593
EP - 642
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 1-2
ER -