Sieving the positive integers by large primes
type: Item
instance of: scholarly article
title: Sieving the positive integers by large primes
zbMATH Open document ID: 0639.10028
DOI: 10.1016/0022-314X(88)90121-7
author: Kevin S. McCurley
author: Daniel Alan Goldston
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(\Psi\) (x,y) be the number of integers \(\leq x\) having no prime factor exceeding y. This function has been extensively studied; e.g. by \textit{A. Hildebrand} [Q. J. Math., Oxf. II. Ser. 36, 57-69 (1985; Zbl 0562.10018)]. Let Q be a subset of primes having relative density \(\delta\) among the primes. In this paper, the authors consider the function \(\Psi\) (x,y,Q), which is the number of positive integers \(\leq x\) that have no prime factors from Q exceeding y. Denote \(u=\log x/\log y\), and let \(\rho_{\delta}\) be the continuous solution of the differential delay equation \(u\rho '_{\delta}(u)=-\delta \rho_{\delta}(u- 1),\rho_{\delta}(u)=1\), \(0\leq u\leq 1\). The authors show that when \(0<\delta <1\), \(\Psi (x,y,Q) \approx x\rho_{\delta}(u).\) The error terms are such that the estimates are uniform for all reasonable values of u and y. By way of contrast, the corresponding result of Hildebrand (ibid.) for \(\Psi\) (x,y) is uniform only for \(y\geq \exp ((\log \log x)^{5/3+\epsilon}).\) One reason for this is that when u is large, \(\Psi\) (x,y) depends strongly on irregularities in the distribution of small primes, but these effects are not felt in \(\Psi\) (x,y,Q).
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N35
zbMATH DE Number: 4042017
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Dickman function
zbMATH Keywords: integers free of large prime divisors
zbMATH Keywords: differential delay equation
zbMATH Keywords: error terms
reviewed by: Sidney W. Graham
MaRDI profile type: Publication
cites work: Q5803354
cites work: Q5514677
cites work: Q5805143
cites work: Q4749890
cites work: Q4765076
cites work: Q4052222
cites work: On the number of positive integers \(\leq x\) and free of prime factors \(>y\)
cites work: INTEGERS FREE OF LARGE PRIME DIVISORS IN SHORT INTERVALS
cites work: On Integers Free of Large Prime Factors
cites work: Q4092051
cites work: Q3998724
cites work: Analysis of a simple factorization algorithm
cites work: Numbers with small prime factors, and the least 𝑘th power non-residue
cites work: Q4059363
cites work: Über die Zahlen, deren Primteiler einer gegebenen Menge angehören
Recommended article: The sum of \(\alpha^{\Omega(n)}\) over integers n\(\leq x\) with all prime factors between \(\alpha\) and y
Recommended article: Q4210651
Recommended article: On Integers Free of Large Prime Factors
Recommended article: Q3023802
Recommended article: Q3029085
Recommended article: On the Local Behavior of Ψ(x, y)
Recommended article: Sieving by large prime factors
Recommended article: Q3220662
Recommended article: A property of the counting function of integers with no large prime factors
Recommended article: Integers with no large prime factors
description: scientific article; zbMATH DE number 4042017