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One-Level Density of Low-lying Zeros of Quadratic and Quartic Hecke $L$-functions

Part of: Zeta and $L$-functions: analytic theory Exponential sums and character sums Algebraic number theory: global fields

Published online by Cambridge University Press:  30 August 2019

Peng Gao  and
Liangyi Zhao
Peng Gao
Affiliation:
School of Mathematics and Systems Science, Beihang University, Beijing 100191, China Email: penggao@buaa.edu.cn
Liangyi Zhao
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia Email: l.zhao@unsw.edu.au
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Abstract

In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.

Keywords

one level densitylow-lying zeroquadratic Hecke characterquartic Hecke characterHecke L-function

MSC classification

Primary: 11L40: Estimates on character sums
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M50: Relations with random matrices 11R16: Cubic and quartic extensions
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 2 , April 2020 , pp. 427 - 454
Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author P. G. was supported in part by NSFC grant 11871082, and author L. Z. was supported by FRG grant PS43707.

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