On the structure of the Selberg class. V: \(1<d<5/3\)
type: Item
instance of: scholarly article
title: On the structure of the Selberg class. V: \(1<d<5/3\)
zbMATH Open document ID: 1034.11051
author: J. Kaczorowski
author: A. Perelli
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: The authors continue their comprehensive study of functions belonging to the Selberg \(\mathcal S\) class of Dirichlet series [see the authors' article for the relevant definitions and properties of \(\mathcal S\), in: Number Theory in Progress, 953--992 (1999; Zbl 0929.11028), see also their works in Acta Math. 182, 207--241 (1999; Zbl 1126.11335), J. Reine Angew. Math. 524, 73--96 (2000; Zbl 0982.11051) and Duke Math. J. 101, 529--554 (2000; Zbl 0997.11072)]. Recall that the extended Selberg class \(\mathcal S^\sharp\) consists of Dirichlet series \(F(s)\) with meromorphic continuation over \(\mathbb C\) such that \((s-1)^mF(s)\) is an entire function of finite order for some integer \(m\geq 0\), which satisfy the functional equation \[ \gamma(s)F(s) = \omega{\bar \gamma}(1-s){\bar F}(1-s), \] where \(| \omega| = 1, {\bar F}(s) = \overline {F({\bar s})}\) and \[ \gamma(s) = Q^s\prod_{j=1}^r\Gamma(\lambda_js+\mu_j),\;Q>0, \,\lambda_j > 0,\,\Re\,\mu_j\geq0. \] If the degree of \(F\in\mathcal S^\sharp\) is defined as \(d_F = 2\sum_{j=1}^r\lambda_j\), then a sensible conjecture asserts that \(d_F\) is always a non-negative integer. The authors make a significant contribution to this subject by proving that there are no functions \(F\) from \(S^\sharp\) satisfying \(1 < d_F < 5/3\). If the order of the pole of \(F(s)\) at \(s=1\) is denoted by \(m_F\), then it is also proved that for \(1 < d < 2\) there exist no \(F\) from \(S^\sharp\) with \(m_F \geq 1\). Furthermore, the authors' arguments show that if the degree conjecture holds for all \(F\in\mathcal S^\sharp\) with \(m_F\geq 1\), then it holds also for all \(F\in\mathcal S^\sharp\). Since it was proved long ago by \textit{H.-E. Richert} [Publ. Inst. Math. Acad. Serbe Sci. 11, 73--124 (1957; Zbl 0082.05802)] that \(\mathcal S^\sharp\) is empty for \(0 < d < 1\), we have now a fairly good knowledge of the structure of \(\mathcal S^\sharp\) for \(d < 5/3\); moreover the authors state that they can extend conditionall…
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1965451
zbMATH Keywords: extended Selberg class
zbMATH Keywords: linear twists
zbMATH Keywords: degree conjecture
zbMATH Keywords: Fox functions
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
full work available at URL: s00222-002-0236-9
OpenAlex ID: W1977737579
DOI: 10.1007/S00222-002-0236-9
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description: scientific article; zbMATH DE number 1965451