Zero-density estimates for Epstein zeta functions of class numbers 2 or 3

Yoonbok Lee

doi:10.4134/JKMS.j160066

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2017; 54(2): 479-491

Online first article January 4, 2017 Printed March 1, 2017

https://doi.org/10.4134/JKMS.j160066

Zero-density estimates for Epstein zeta functions of class numbers 2 or 3

Yoonbok Lee

Incheon National University

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Abstract

We investigate the zeros of Epstein zeta functions associated with positive definite quadratic forms with rational coefficients in the vertical strip $ \sigma_1 < \Re s < \sigma_2 $, where $ 1/2 < \sigma_1 < \sigma_2 < 1 $. When the class number $h$ of the quadratic form is bigger than 1, Voronin gave a lower bound and Lee gave an asymptotic formula for the number of zeros. Recently Gonek and Lee improved their results by providing a new upper bound for the error term when $h>3$. In this paper, we consider the cases $h=2,3$ and provide an upper bound for the error term, smaller than the one for the case $h>3$.

Keywords: Epstein zeta function, zero density, Hecke $L$-function

MSC numbers: 11M26, 11M41