On the value-distribution of Epstein zeta-functions

Abstract

We investigate the value-distribution of Epstein zeta-functions $\zeta(s;{\mathcal Q})$, where ${\mathcal Q}$ is a positive definite quadratic form in $n$ variables. We prove an asymptotic formula for the number of $c$-values, i.e., the roots of the equation $\zeta(s;{\mathcal Q})=c$, where $c$ is any fixed complex number. Moreover, we show that, in general, these $c$-values are asymmetrically distributed with respect to the critical line $\operatorname{Re} s=\frac{n}{4}$. This complements previous results on the zero-distribution [30].