On the small sieve. I. Sifting by primes
type: Item
instance of: scholarly article
title: On the small sieve. I. Sifting by primes
zbMATH Open document ID: 0435.10028
DOI: 10.1016/0022-314X(80)90032-3
author: Imre Z. Ruzsa
author: P. Erdős
published in: Journal of Number Theory
publication date: 1980-01-01T00:00:00Z
review text: The authors continue some investigations on what the reviewer has called the Erdős-Szemerédi sieve. Let \(A\) be a set of natural numbers not containing 1. Let \(F(x,A)\) denote the number of natural numbers \(n\leq x\), not divisible by any element of \(A\). Let \(K> 0\) be any constant. Let \(P\) run over all possible sets of primes the sum of whose reciprocals do not exceed \(K\). Put \(G(x,K)=\min F(x,P)\). The authors prove that \[ G(x,K)\geq x(\exp\exp (cK))^{-1}, \] where \(x\geq 2\) and \(c\) is an absolute positive constant. (The proof involves a curious induction procedure which they call real type induction). They have also other results. For example if \(P\) is contained in \([2,x^{1-\delta}]\) then \(G(x,K)\geq c\), \(\delta e^{-K}x\), where \(\delta> 0\) is arbitrary and \(c_1\) is an absolute positive constant. They also study \(\min F(x,A)\) where \(A\) ranges over more general sets of integers.
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 3677949
zbMATH Keywords: small sieve
zbMATH Keywords: sifting by primes
zbMATH Keywords: coprime sifting set
MaRDI profile type: Publication
Wikidata QID: Q105942337
cites work: Generalization of an inequality of Heilbronn and Rohrbach
cites work: Q5803354
cites work: Q5515488
cites work: Q5766164
cites work: Probabilistic Generalization of a Number-Theoretical Inequality
cites work: Q3279332
full work available at URL: 0022-314x(80)90032-3
OpenAlex ID: W2140029208
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description: scientific article; zbMATH DE number 3677949