The prime number theorem for Beurling's generalized numbers. New cases
type: Item
instance of: scholarly article
title: The prime number theorem for Beurling's generalized numbers. New cases
zbMATH Open document ID: 1300.11099
author: J. Vindas
author: Jan-Christoph Schlage-Puchta
published in: Acta Arithmetica
publication date: 2012-04-13T00:00:00Z
full work available at URL: LU-3023509
Mathematics Subject Classification ID: 11N80
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11M45
Mathematics Subject Classification ID: 40E05
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 46F10
zbMATH DE Number: 6023952
zbMATH Keywords: Beurling generalized numbers
zbMATH Keywords: prime number theorem
zbMATH Keywords: Tauberian theorems
MaRDI profile type: Publication
OpenAlex ID: W2039512837
DOI: 10.4064/AA153-3-5
review text: \textit{A. Beurling} [Acta Math. 68, 255--291 (1937; Zbl 0017.29604)] has defined generalized prime numbers \(p_k\) and integers \(n_k\) \((k\geq 1)\). He proved the prime number theorem (PNT) NEWLINE\[NEWLINE\pi(x):= \sum_{p_k<x} 1\sim{x\over\log x}NEWLINE\]NEWLINE under the condition NEWLINE\[NEWLINEN(x):= \sum_{n_k<x} 1= ax+ O\Biggl({x\over\log^\gamma x}\Biggr),NEWLINE\]NEWLINE where \(a>0\), \(\gamma>{3\over 2}\). The authors extend Beurling's theorem by replacing this condition by an asymptotic in Cesàro sense NEWLINE\[NEWLINE\int^x_1 {N(t)-at\over t} \Biggl(1-{t\over x}\Biggr)^m \,dt= O\Biggl({x\over\log^\gamma x}\Biggr).NEWLINE\]NEWLINE The proof works by Tauberian theorems and is close to Landau-Ikehara's approach to the PNT. The authors construct a number system such that \(\gamma={3\over 2}\) and the PNT does not hold. Therefore, the condition \(\gamma>{3\over 2}\) is sharp.
reviewed by: Jürgen Spilker
description: scientific article; zbMATH DE number 6023952