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

Mathematics

Antanas Laurinčikas¹  Renata Macaitienė² 

[1] Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania;Institute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto Str. 84, LT-76352 Šiauliai, Lithuania;

关键词: Epstein zeta-function;    limit theorem;    weak convergence;    Haar measure;   

DOI  :  10.3390/math10122042

来源: DOAJ

【 摘 要 】

Let Q be a positive defined $n\times n$ matrix and $Q\left[\underline{x}\right]={\underline{x}}^{\mathrm{T}}Q\underline{x}$. The Epstein zeta-function $\zeta (s;Q)$, $s=\sigma +it$, is defined, for $\sigma >\frac{n}{2}$, by the series $\zeta (s;Q)={\sum }_{\underline{x}\in {\mathbb{Z}}^n\backslash \left\{\underline{0}\right\}}$${\left(Q\left[\underline{x}\right]\right)}^{-s},$ and is meromorphically continued on the whole complex plane. Suppose that $n⩾4$ is even and $\phi \left(t\right)$ is a differentiable function with a monotonic derivative. In the paper, it is proved that $\frac{1}{T}\mathrm{meas}\left\{t\in [0,T]:\zeta (\sigma +i\phi (t);Q)\in A\right\}$, $A\in \mathcal{B}\left(\mathbb{C}\right),$ converges weakly to an explicitly given probability measure on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$ as $T\to \infty$.

【 授权许可】

Unknown