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 $\epsilon >0$, we determine the limit distribution of the random function $c\mapsto V_n^{-2c}{E}_n(\cdot ,cn)$, $c\in [1/4+\epsilon ,1/2-\epsilon ]$. After compensating for the pole at $c=1/2$, we even obtain a limit result on the whole interval $[1/4+\epsilon ,1/2]$, and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function $h_n\left(L\right)$ of the flat torus ${\mathbb{R}}^n/L$: the random variable $n\left\{h_n\right(L)-(log\left(4\pi \right)-\gamma +1\left)\right\}+logn$ has a limit distribution as $n\to \infty$, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice $L\subset {\mathbb{R}}^n$ for which $E_n(L,s)$ has no zeros in $(0,\infty )$.