On Beurling's prime number theorem
type: Item
instance of: scholarly article
title: On Beurling's prime number theorem
zbMATH Open document ID: 0826.11044
DOI: 10.1007/BF01876339
published in: Periodica Mathematica Hungarica
publication date: 1995-11-28T00:00:00Z
review text: The author studies a ``Beurling'' zeta function \[ \zeta (s)= \exp \Biggl( \int_1^\infty x^{-s} d\Pi (x) \Biggr)= 1+ \int_1^\infty x^{-s} dN(x) \] for \(\text{Re} (s)>1\), analytic in \(s\) for \(\text{Re} (s)>1\), where \(\Pi\) is a nondecreasing real function with \(\Pi (1) =0\), and \(N\) is a nondecreasing function with \(N(1) =0\), determined by \(\Pi\). Subject to varying asymptotic assumptions on \(N(x)\) as \(x\to \infty\), \textit{A. Beurling} [Acta Math. 68, 255-291 (1937; Zbl 0017.29604)] derived corresponding asymptotic conclusions about the function \(\Pi (x)\) as \(x\to \infty\). The starting point for such investigations was that sufficiently strong assumptions about \(N(x)\) imply analogues of the classical prime number theorem. The present author continues this line of investigation on the basis of more general assumptions about \(N(x)\). As with Beurling and other, later authors, the emphasis of the work is on the analytic techniques and results which varying analytic hypotheses lead to.
Mathematics Subject Classification ID: 11N80
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 698624
zbMATH Keywords: Beurling zeta function
zbMATH Keywords: nondecreasing real function
zbMATH Keywords: asymptotic conclusions
zbMATH Keywords: analytic techniques
author: Szilárd Gy. Révész
reviewed by: John Knopfmacher
MaRDI profile type: Publication
cites work: Q5619868
cites work: Q5849097
cites work: Analyse de la loi asymptotique de la distribution des nombres premiers généralises
cites work: The prime number theorem for Beurling's generalized numbers
cites work: Q3978166
full work available at URL: https://doi.org/10.1007/bf01876339
OpenAlex ID: W1984726203
description: scientific article; zbMATH DE number 698624