On the value distribution of the Epstein zeta function in the critical strip

TL;DR

This paper investigates the distribution of values of the Epstein zeta function for high-dimensional random lattices, establishing explicit limit distributions and strengthening known results about lattice height functions.

Contribution

It provides explicit limit distributions for the Epstein zeta function in the critical strip for large dimensions, extending previous results and addressing open questions about lattice zeros.

Findings

Limit distribution of $V_n^{-2c}E_n(ullet,cn)$ for fixed $c$ as $n oty$

Limit distribution of the height function $h_n(L)$ with explicit form

Discussion on existence of lattices with no zeros of $E_n(L,s)$ in $(0,ty)$

Abstract

We study the value distribution of the Epstein zeta function $E_n(L,s)$ for $0<s<\frac{n}{2}$ and a random lattice $L$ of large dimension $n$. For any fixed $c\in(1/4,1/2)$ and $n\to\infty$, we prove that the random variable $V_n^{-2c}E_n(\cdot,cn)$ has a limit distribution, which we give explicitly (here $V_n$ is the volume of the $n$-dimensional unit ball). More generally, for any fixed $\ve>0$ we determine the limit distribution of the random function $c\mapsto V_n^{-2c}E_n(\cdot,cn)$, $c\in[1/4 +\ve, 1/2-\ve]$. After compensating for the pole at $c=\frac12$ we even obtain a limit result on the whole interval $[\frac14+\ve,\frac12]$, and as a special case we deduce the following strengthening of a result by Sarnak and Str\"ombergsson concerning the height function $h_n(L)$ of the flat torus $\R^n/L$: The random variable $n\big\{h_n(L)-(\log(4\pi)-\gamma+1)\big\}+\log n$ has a limit distribution as $n\to\infty$, which we give explicitly. Finally we discuss a question posed by Sarnak and Str\"ombergsson as to whether there exists a lattice $L\subset\R^n$ for which $E_n(L,s)$ has no zeros in $(0,\infty)$.