Beurling primes with RH and Beurling primes with large oscillation
type: Item
instance of: scholarly article
title: Beurling primes with RH and Beurling primes with large oscillation
zbMATH Open document ID: 1121.11061
author: Wenbin Zhang
published in: Mathematische Annalen
publication date: 2007-04-19T00:00:00Z
review text: From the author's abstract: Two Beurling generalized number systems, both with \(N(x)=kx+O(x^{1/2}\exp\{c(\log x)^{2/3}\})\) and \(k>0\) are constructed. The associated zeta function of the first satisfies the RH (Riemann Hypothesis) and ist prime counting function satisfies \(\pi(x)=li(x)+O(x^{1/2}\). The associated zeta function of the second has infinitely many zeros on the curve \(\sigma=1-1/\log t\) and no zeros to the right of the curve and the Chebyshev function \(\psi(x)\) of its primes satisfies \[ \begin{aligned} &\limsup\bigl(\psi(x)-x\bigr)/\bigl(x\exp\{-2\sqrt{\log x}\,\}\bigr)=2 \qquad \text{and}\\ &\liminf\bigl(\psi(x)-x\bigr)/\bigl(x\exp\{-2\sqrt{\log x}\,\}\bigr)=-2. \end{aligned} \] A sharpened form of the Diamond-Montgomery-Vorhauer random approximation and elements of analytic number theory are used in the construction.
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11N80
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 5145030
zbMATH Keywords: Beurling primes
MaRDI profile type: Publication
full work available at URL: s00208-006-0051-5
OpenAlex ID: W2091424942
cites work: Q5619868
cites work: Q4829664
cites work: Analyse de la loi asymptotique de la distribution des nombres premiers généralises
cites work: Beurling primes with large oscillation
cites work: Q3874481
DOI: 10.1007/S00208-006-0051-5
description: scientific article; zbMATH DE number 5145030