A Bohr-Jessen type theorem for the Epstein zeta-function
It is known that the Epstein zeta-function \(\zeta (s;Q),s=\sigma +it\), is defined for \(\sigma >n/2\) by the series \[ \zeta (s;Q)=\sum_{\underline{x}\in \mathbb{Z}^{n}-\{0\}}\frac{1}{(Q[\underline{x}])^{s}}, \] where \(Q\) is a positive definite quadratic \(n\times n\) matrix and \(Q[\underline{x}])=\underline{x}^{T}Q\underline{x}\), and is meromorphically continued to the whole complex plane. In this paper, the authors consider a Bohr-Jessen type theorem for the Epstein zeta-function \(\zeta (s;Q)\) and prove that for \(\sigma >(n-1)/2,T\rightarrow \infty \), a limit theorem on the weak convergence of \[ \frac{1}{T}\text{meas}\{t\in [ 0,T]:\zeta (\sigma +it;Q)\in A\}, \] where \(A\in \mathcal{B}(\mathbb{C})\).
scientific article; zbMATH DE number 6995204