On the zeros of Epstein zeta functions

2012/04/27 by Yoonbok Lee, Lee, Yoonbok
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1204.6297

Recently revised version

arxiv created 2012/04/27 · arxiv updated 2012/04/30

Abstract

We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.

Related

• Zero-density estimates for Epstein zeta functions
• On the zeros of Epstein zeta functions near the critical line
• The Epstein zeta-function contains a positive proportion of non-trivial zeros on the critical line
• A note on the gaps between zeros of Epstein's zeta-functions on the critical line
• On the Epstein zeta function and the zeros of a class of Dirichlet series