On the value distribution and moments of the Epstein zeta function to the right of the critical strip

TL;DR

This paper investigates the distribution and moments of the Epstein zeta function for large dimensions, revealing its asymptotic behavior and limit distributions for fixed and varying parameters.

Contribution

It provides the first detailed analysis of the value distribution and moments of the Epstein zeta function beyond the critical strip for high dimensions.

Findings

Asymptotic distribution of $E_n(L,cn)$ as $n oty$

Limit distribution of the random function $c o E_n(ullet,cn)$ for $c$ in a fixed interval

Explicit characterization of moments for large $n$

Abstract

We study the Epstein zeta function $E_n(L,s)$ for $s>\frac{n}{2}$ and determine for fixed $c>\frac{1}{2}$ the value distribution and moments of $E_n(\cdot,cn)$ (suitably normalized) as $n\to\infty$. We further discuss the random function $c\mapsto E_n(\cdot,cn)$ for $c\in[A,B]$ with $\frac{1}{2}<A<B$ and determine its limit distribution as $n\to\infty$.