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

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 $(0,\infty)$.