A PNT equivalence for Beurling numbers
type: Item
instance of: scholarly article
title: A PNT equivalence for Beurling numbers
zbMATH Open document ID: 1318.11124
author: Harold G. Diamond
author: Wenbin Zhang
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2012-08-31T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1340628404
review text: The objective of the paper is to investigate equivalences between the prime number theorem and the sharp Merten's formula for Beurling generalized numbers introduced in [\textit{A. Beurling}, Acta Math. 68, 255--291 (1937; Zbl 0017.29604)]. Beurling generalized numbers form a multiplicative semigroup \(\mathcal{N}\subset\mathbb{R}\) generated by Beurling generalized prime system \(\mathcal{P}\) formed by an unbounded sequence of real number \(1<p_1\leq p_2\leq\dots\). Write \(N_{\mathcal{P}}(x)\) for the number of such generalized integers in \([1,x]\), and define \(\pi_{\mathcal{P}}(x)\) and \(\psi_{\mathcal{P}}(x)\) in analogy to the definitions of these function for the usual set of primes. The main results of the paper says: Suppose that the PNT holds and that for some \(c>0\) we have \(|N_{\mathcal{P}}(x)-cx|\leq xD(x)\) for \(x\geq1\), where \(D\) is right continuous, monotone decreasing, and satisfies \(\int_1^\infty x^{-1}D(x)\,dx<\infty\). Then there follows the sharp Mertens relation \(\int_1^x (1/t)\,d\psi_{\mathcal{P}}(t)=\log x+c_1+o(1)\) for some constant \(c_1\). The authors show that this sharp Merten's relation implies the PNT, the PNT alone is not sufficient for the validity of the sharp Merten's relation, however. For its validity even PNT with the hypotheses \(N_{\mathcal{P}}=cx+O(x/\log x)\) is not sufficient, as shown by the authors in terms of a continuous example of Beurling generalized numbers.
reviewed by: Štefan Porubský
Mathematics Subject Classification ID: 11N80
zbMATH DE Number: 6074841
zbMATH Keywords: Beurling generalized numbers
zbMATH Keywords: sharp Mertens formula
zbMATH Keywords: prime number theorem
zbMATH Keywords: equivalent relations
MaRDI profile type: Publication
OpenAlex ID: W2030860563
cites work: Q5619868
cites work: Analyse de la loi asymptotique de la distribution des nombres premiers généralises
cites work: Asymptotic distribution of Beurling's generalized integers
cites work: Q3423329
DOI: 10.7169/FACM/2012.46.2.7
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description: scientific article; zbMATH DE number 6074841