Unique factorization results for semigroups of \(L\)-functions
type: Item
instance of: scholarly article
title: Unique factorization results for semigroups of \(L\)-functions
zbMATH Open document ID: 1163.11062
author: J. Kaczorowski
author: Giuseppe Molteni
author: A. Perelli
published in: Mathematische Annalen
publication date: 2008-06-04T00:00:00Z
review text: The extended Selberg class \(S^{\#}\) is defined as the class of non-identically vanishing Dirichlet series \[ F(s)=\sum_{n=1}^{\infty}a_F(n)n^{-s},\quad s=\sigma+it, \] absolutely convergent for \(\sigma>1\). The function \(F(s)\) satisfies a functional equation \[ \Phi(s)=\omega {\overline {\Phi(\overline{1-s})}}, \] where \(|\omega|=1\) and \[ \Phi(s)=Q^s\prod_{j=1}^{r}\Gamma(\lambda_j s+\mu_j)F(s) \] with \(r \geq 0\), \(Q>0\), \(\lambda_j>0\) and \(\Re \mu_j \geq 0\) (\(r=0\) meaning that there are no \(\Gamma\)-factors). The degree of a function \(F \in S^{\#}\) is defined by \[ d_F=2\sum_{j=1}^{r}\lambda_j. \] In the paper, the authors prove that the set of functions \(F \in S^{\#}\) with degree 0 has unique factoriation. Also, they consider the multiplicative semigroups generated by the functions \(F \in S^{\#}\) with degree 1, and prove that such set has unique factorization. The authors consider some extensions of above mentioned results leading to similar results involving certain classical \(L\)-functions.
reviewed by: Roma Kačinskaitė
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 5284754
zbMATH Keywords: Selberg class
zbMATH Keywords: \(L\)-functions
MaRDI profile type: Publication
full work available at URL: s00208-007-0197-9
OpenAlex ID: W2036776808
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DOI: 10.1007/S00208-007-0197-9
description: scientific article; zbMATH DE number 5284754