Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals - Balčiūnas, Aidas; Garbaliauskienė, Virginija; Macaitienė, Renata (Mathematical Modelling and Analysis, 2026)
Let $Q$ be a positive definite $n × n$ matrix, $n \in 2\mathbb{N}$, $n \geqslant4$. The Epstein zeta-function $ζ(s; Q)$, defined for $Re\,s > \tfrac{n}{2}$, is given by $ζ(s; Q) = \sum_{\underline{x} \in \mathbb{Z}^n \setminus \{\underline{0}\}} (\underline{x}^T Q{\underline{x}})^{-s},$ and has a meromorphic continuation to the whole complex plane. Let $T^{{27}/{82}} \leqslant H \leqslant T^{{1}/{2}}$. In this paper, we prove a limit theorem on weak convergence for $\frac{1}{H} meas\left\{t \in [T, T+H]: ζ(σ + it; Q) \in A \right\},\; A \in \mathcal{B}(\mathbb{C}),$ as $T→∞$, where $\mathcal{B}(\mathbb{C})$ is the Borel $σ$-algebra on $\mathbb{C}$. The limit measure is explicitly given. The result extends a known theorem obtained for the interval $[0, T]$.