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A zero density estimate and fractional imaginary parts of zeros for $\textrm{GL}_2$ L-functions

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  28 December 2022

OLIVIA BECKWITH ,
DI LIU ,
JESSE THORNER  and
ALEXANDRU ZAHARESCU
OLIVIA BECKWITH
Affiliation:
Mathematics Department, Tulane University, New Orleans, 6823 St. Charles Ave., LA 70118, U.S.A. e-mail: obeckwith@tulane.edu
DI LIU
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A. e-mails: dil4@illinois.edu, jesse.thorner@gmail.com
JESSE THORNER
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A. e-mails: dil4@illinois.edu, jesse.thorner@gmail.com
ALEXANDRU ZAHARESCU
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A. and Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania. e-mail: zaharesc@illinois.edu
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Abstract

We prove an analogue of Selberg’s zero density estimate for $\zeta(s)$ that holds for any $\textrm{GL}_2$ L-function. We use this estimate to study the distribution of the vector of fractional parts of $\gamma\boldsymbol{\alpha}$ , where $\boldsymbol{\alpha}\in\mathbb{R}^n$ is fixed and $\gamma$ varies over the imaginary parts of the nontrivial zeros of a $\textrm{GL}_2$ L-function.

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 3 , May 2023 , pp. 605 - 630
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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