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ON SIMPLE ZEROS OF THE DEDEKIND ZETA-FUNCTION OF A QUADRATIC NUMBER FIELD

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

Published online by Cambridge University Press:  22 May 2019

Xiaosheng Wu  and
Lilu Zhao
Xiaosheng Wu
Affiliation:
School of Mathematics, Hefei University of Technology, Hefei 230009, China email xswu@amss.ac.cn
Lilu Zhao
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China email zhaolilu@sdu.edu.cn
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Abstract

We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle $\{\unicode[STIX]{x1D70E}+\text{i}t:0<\unicode[STIX]{x1D70E}<1,0<t<T\}$. We prove that such a number exceeds $T^{6/7-\unicode[STIX]{x1D700}}$ if $T$ is sufficiently large. This improves upon the classical lower bound $T^{6/11}$ established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$ 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11R42: Zeta functions and $L$-functions of number fields
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 851 - 861
Copyright
Copyright © University College London 2019 

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