On some generalized integers
type: Item
instance of: scholarly article
title: On some generalized integers
zbMATH Open document ID: 0883.11044
DOI: 10.1007/BF02473979
published in: Lithuanian Mathematical Journal
publication date: 1997-10-29T00:00:00Z
review text: Let \((q_i)\) be a sequence of generalized primes, i.e. \( q_i\) are real numbers and \( 1 = q_0 <q_1\leq q_2 \leq \cdots\), \(q_i \to \infty\), and \( (n_i)_{i \in \mathbb N_0} \) be the associated set of Beurling integers. Define for \(x >0\) the counting functions \[ B(x) :=\sum_{n_i \leq x} 1; \quad P(x) := \sum_{q_i\leq x}1 . \] \textit{A. Beurling} [Acta Math. 68, 255-291 (1937; Zbl 0017.29604)] started the study of such number systems. Essentially, two problems were investigated: 1) to deduce an asymptotic formula for \(P(x)\) and, in particular, an analogon of the prime number theorem \(P(x) \sim x/\log x\) from asymptotics of \(B(x)\), and 2) in the other direction to prove an asymptotic formula for \(B(x)\) under certain assumptions on the behavior of the generalized primes. In the paper under review the author considers a problem of the second kind: for given \(r>0\) define generalized primes \(q_i\) by \(q_i:= p_{m+i}-r\), \(i=1,2,\ldots\), where \(m=\pi(r)+1\) and \(p_n\) is the ordinary \(n\)-th rational prime number. Then there exists an explicitly given constant \(b(r)>0\) and a remainder \(R(x)\) of order roughly \(O(x\exp(-\log^{1/2}x))\) such that \[ B(x)=b(r)x+R(x). \] The case \(r=1\) is connected with the well studied sum \(\sum_{\varphi(n)\leq x}1\), where \(\varphi\) denotes Euler's function. In the proofs the author uses the classical methods of contour integration. As he remarked, his remainder term (in the case \(r=1\)) is not as sharp as a result which M. Balazard obtained quite recently.
Mathematics Subject Classification ID: 11N80
zbMATH DE Number: 1059803
zbMATH Keywords: Beurling numbers
zbMATH Keywords: generalized integers
zbMATH Keywords: generalized primes
author: Eugenijus Stankus
reviewed by: Helmut Müller
MaRDI profile type: Publication
cites work: The distribution of values of the Euler function
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cites work: Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen \(\sigma = 1\)
cites work: The analytic continuation of Euler products with applications to asymptotic formulae
cites work: Evaluation effective du nombre d'entiers n tels que φ(n) ≤ x
full work available at URL: https://doi.org/10.1007/bf02473979
OpenAlex ID: W2081011449
description: scientific article; zbMATH DE number 1059803