On the structure of the Selberg class. I: \(0\leq d\leq 1\).
type: Item
instance of: scholarly article
title: On the structure of the Selberg class. I: \(0\leq d\leq 1\).
zbMATH Open document ID: 1126.11335
DOI: 10.1007/BF02392574
author: J. Kaczorowski
author: A. Perelli
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: \textit{A. Selberg} [Proceedings of the Amalfi conference on analytic number theory, held at Maiori, Amalfi, Italy, from 25 to 29 September, 1989. Salerno: Universitá di Salerno, 367--385 (1992; Zbl 0787.11037)] defined a class of Dirichlet series which includes the Riemann zeta-function, Dirichlet \(L\)-functions, \(L\)-functions attached to modular forms (at least conjecturally), and essentially all Dirichlet series where one might expect a Riemann hypothesis. Namely, functions in this class have precise global analytic behavior, functional equations, Euler products, and tight bounds on the coefficients. A study of this class and Selberg's conjectures has been taken up by other authors (notably Conrey, Ghosh, Bochner, and Murty; references are in the paper under review). A fundamental problem in this arena is the characterization of the admissible functional equations and of the so-called primitive functions in the class. In this paper, the authors take up some of these questions and provide complete and unconditional results for one of the more interesting subclasses (that containing the Riemann zeta-function, Dirichlet \(L\)-functions and Dirichlet polynomials). They determine that these are essentially all the functions in this subclass. This includes and extends results of Conrey-Ghosh and Bochner. The methodology involves some intricate analysis using incomplete Fox hypergeometric functions. But the most interesting aspect of their methodology is the resurrection of Linnik's method of additive twists. The (perhaps a priori more natural) multiplicative twists depend on the Euler product condition of the class definition. This additive approach has the side consequence that the Euler product assumptions are not required; in fact these properties are deduced from the final characterization. This approach also admits a vector-space structure on the subclass; the authors prove dimension results for this space, as well.
reviewed by: James Lee Hafner
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1541208
MaRDI profile type: Publication
cites work: On Riemann's function equation with multiple gamma factors
cites work: Q5338397
cites work: On the Selberg class of Dirichlet series: Small degrees
cites work: Q5820721
cites work: On the structure of the Selberg class, II: invariants and conjectures
cites work: Q5791262
cites work: Selberg’s conjectures and Artin 𝐿-functions
cites work: Q3735790
description: scientific article; zbMATH DE number 1541208