A remark on solutions of functional equations of Riemann's type
type: Item
instance of: scholarly article
title: A remark on solutions of functional equations of Riemann's type
zbMATH Open document ID: 1065.11071
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2005-03-30T00:00:00Z
review text: Let \(\mathcal C\) be a class of generalized Dirichlet series \[ F(s)=\sum_{n=1}^\infty\frac{a(n)}{l_n^s} \] absolutely convergent for \(\sigma\) sufficiently large, where \((l_n)\) is an increasing sequence of positive real numbers tending to \(\infty\), such that \(F(s)\) is an entire function of finite order. For given \(\omega\in\mathbb{C}\) with \(| \omega| =1\), \(Q>0\), \(\lambda=(\lambda_1, \dots, \lambda_r)\) with \(\lambda_j>0\) and \(\mu=(\mu_1,\dots, \mu_r)\) with \(\Re \mu_j\geq0\) the authors consider the functional equation \[ \Phi(s)=\omega\overline{\Phi(1-s)},\tag{\(*\)} \] where \(\overline{f}(s)=\overline{f(\overline{s})}\) and \[ \Phi(s)=Q^s\prod_{j=1}^r\Gamma(\lambda_j s+\mu_j) F(s). \] Denote by \(D(\lambda, \mu, Q, \omega)\) the real vector space of solutions \(F\in\mathcal C\) of functional equation \((*)\). Then the authors prove that \(D(\lambda, \mu, Q, \omega)\) has an uncountable basis. This result is in contrast with the situation in the extended Selberg class (special case of \(\mathcal C\), with \((l_n)=\mathbb{N}\)) where there exist functional equations without non-trivial solutions, see \textit{J. Kaczorowski} and \textit{A. Perelli} [Invent. Math. 150, No. 3, 485--516 (2002; Zbl 1034.11051)]. The proof is based on Hecke's theory of Dirichlet series associated with modular forms.
reviewed by: Ramūnas Garunkštis
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 2149316
zbMATH Keywords: Riemann's functional equation
zbMATH Keywords: Selberg class
zbMATH Keywords: Dirichlet series
zbMATH Keywords: Hecke modular forms
author: J. Kaczorowski
author: A. Perelli
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2149316