A Bohr–Jessen Type Theorem for the Epstein Zeta-Function: II

Abstract

Let Q be a positive definite quadratic matrix $n\times n$, and $Q[{\underline{x}}]= {\underline{x}}^{\mathrm {T}} Q {\underline{x}}$. The Epstein zeta-function $\zeta (s; Q)$, $s=\sigma +it$, is defined, for $\sigma >\tfrac{n}{2}$, by the series

$$\begin{aligned} \zeta (s; Q)=\sum _{{\underline{x}}\in {\mathbb {Z}}^n\setminus \{{\underline{0}}\}} (Q[{\underline{x}}])^{-s}, \end{aligned}$$

and is meromorphically continued to the whole complex plane. In the paper, we prove, for $\sigma >\tfrac{n-1}{2}$, a discrete limit theorem on the weak convergence, as $N\rightarrow \infty$, of

$$\begin{aligned} \frac{1}{N+1} \# \left\{ 0\leqslant k\leqslant N: \zeta (\sigma +ikh; Q)\in A\right\} , \quad A\in {\mathcal {B}}({\mathbb {C}}), \end{aligned}$$

for every fixed $h>0$. Two cases depending on the arithmetic nature of h are investigated, and the explicit form of the limit measure is given.