On the zeros of Epstein zeta functions near the critical line

2018/11/05 by Yoonbok Lee, Lee, Yoonbok
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1811.01613

arxiv created 2018/11/05 · arxiv updated 2018/11/06

Abstract

Let Q be a positive definite quadratic form with integral coefficients and let E(s,Q) be the Epstein zeta function associated with Q. Assume that the class number of Q is bigger than 1. Then we estimate the number of zeros of E(s,Q) in the region \Re s > σ_T ( θ) := 1/2 + ( log T)^{- θ} and T < \Im s < 2T, to provide its asymptotic formula for fixed 0 < θ< 1 conditionally. Moreover, it is unconditional if the class number of Q is 2 or 3 and 0 < θ< 1/13.

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