Extension of analytic number theory and the theory of regularized harmonic series from Dirichlet series to Bessel series
type: Item
instance of: scholarly article
title: Extension of analytic number theory and the theory of regularized harmonic series from Dirichlet series to Bessel series
zbMATH Open document ID: 0871.11058
DOI: 10.1007/BF01445243
author: Serge Lang
author: Jay Jorgenson
published in: Mathematische Annalen
publication date: 1996-11-25T00:00:00Z
full work available at URL: https://eudml.org/doc/165446
review text: Zeta functions (Dirichlet series) are connected to theta functions via the Mellin integral transform. This classical interplay of theta and zeta functions was already used by Riemann to continue the zeta function to the plane. In many aspects of analytical number theory only the logarithmic derivatives of the zeta functions occur, giving rise to an additive theory, which also may be applied to functions which are not logarithmic derivatives. One possible extension, which is the concern of the present paper, is the extension to Bessel series instead of Dirichlet series. In a far reaching program the authors aim at extending the analytical number theory from Dirichlet series to Bessel series. In the present paper they generalize Cramér's theorem, which gives an analytic continuation of the theta series attached to the nontrivial zeros of the Riemann zeta-function. They further give ``explicit formulas'' for Bessel series. In the last sections they include examples from spectral theory and differential geometry which indicate the necessity of using Bessel functions instead of Dirichlet series.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 30B50
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 33C60
Mathematics Subject Classification ID: 35P99
Mathematics Subject Classification ID: 35S99
Mathematics Subject Classification ID: 42A99
zbMATH DE Number: 931993
zbMATH Keywords: Bessel series
zbMATH Keywords: Cramér's theorem
zbMATH Keywords: theta series
reviewed by: Anton Deitmar
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 931993