On second moment of Selberg zeta-function for \(\sigma =1\)
type: Item
instance of: scholarly article
title: On second moment of Selberg zeta-function for \(\sigma =1\)
zbMATH Open document ID: 1476.11118
author: Paulius Drungilas
author: Ramūnas Garunkštis
author: A. Novikas
published in: Results in Mathematics
publication date: 2021-08-19T00:00:00Z
review text: For the Selberg zeta function \(Z(s)\) of the modular group, the authors first prove \(Z^{\pm1}(1+it)\ll\log T\) as \(T\to\infty\) and that both the second moment and its reciprocal on \(\Re(s)=1\) exist satisfying \[ \frac1T\int_1^T|Z(1+it)|^{\pm2}dt \ll\frac{\log^2T}{(\log\log T)^{2-\varepsilon}} \] for any \(\varepsilon>0\) as \(T\to\infty\). Then by using this theorem they deduce \[ \lim_{T\to\infty}\frac1{2T}\int_{-T}^T|Z(1+it)|^2dt =\sum_{n=1}^\infty\frac{a_n^2}{y_n^2} \] and that \[ \lim_{T\to\infty}\frac1{T}\int_{1}^T|Z(1+it)|^{-2}dt =\sum_{n=1}^\infty\frac{b_n^2}{y_n^2}, \] where the Dirichlet series expression of the Selberg zeta function and its reciprocal are denoted by \[ Z(s)=\sum_{n=1}^\infty\frac{a_n}{y_n^s}\quad(\Re(s)>1) \] and \[ Z^{-1}(s)=\sum_{n=1}^\infty\frac{b_n}{y_n^s}\quad(\Re(s)>1) \] with \(1=y_1<y_2<y_3\cdots\).
reviewed by: Shin-ya Koyama
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 7383834
zbMATH Keywords: Selberg zeta function
zbMATH Keywords: second moment
zbMATH Keywords: mean value theorems
zbMATH Keywords: Beurling zeta function
MaRDI profile type: Publication
full work available at URL: s00025-021-01492-5
OpenAlex ID: W3195979152
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DOI: 10.1007/S00025-021-01492-5
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description: scientific article; zbMATH DE number 7383834