| Title |
Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals |
| Authors |
Balčiūnas, Aidas ; Garbaliauskienė, Virginija ; Macaitienė, Renata |
| DOI |
10.3846/mma.2026.24285 |
| Full Text |
|
| Is Part of |
Mathematical modelling and analysis.. Vilnius : Vilnius Gediminas Technical University. 2026, vol. 31, no. 2, p. 288-302.. ISSN 1392-6292. eISSN 1648-3510 |
| Keywords [eng] |
Epstein zeta-function ; probability Haar measure ; limit theorem ; weak convergence. |
| Abstract [eng] |
Let $Q$ be a positive definite $n \times n$ matrix, $n \in 2\mathbb{N}$, $n \geqslant4$. The Epstein zeta-function $\zeta(s; Q)$, defined for $\mathrm{Re}\,s > \tfrac{n}{2}$, is given by $\zeta(s; Q) = \sum_{\underline{x} \in \mathbb{Z}^n \setminus \{\underline{0}\}} (\underline{x}^T Q{\underline{x}})^{-s},$ 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 $\frac{1}{H} \mathrm{meas}\left\{t \in [T, T+H]: \zeta(\sigma + it; Q) \in A \right\},\; A \in \mathcal{B}(\mathbb{C}),$ as $T\to\infty$, where $\mathcal{B}(\mathbb{C})$ is the Borel $\sigma$-algebra on $\mathbb{C}$. The limit measure is explicitly given. The result extends a known theorem obtained for the interval $[0, T]$. |
| Published |
Vilnius : Vilnius Gediminas Technical University |
| Type |
Journal article |
| Language |
English |
| Publication date |
2026 |
| CC license |
|
