Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  435.10028
Autor:  Erdös, Paul; Ruzsa, I.Z.
Title:  On the small sieve. I. Sifting by primes. (In English)
Source:  J. Number Theory 12, 385-394 (1980).
Review:  The authors continue some investigations on what the reviewer has called the Erdös-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) = max 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,x1-\delta] then G(x,K) \geq c, \delta e-Kx, where \delta > 0 is arbitrary and c1 is an absolute positive constant. They also study max F(x,A) where A ranges over more general sets of integers.
Reviewer:  K.Ramachandra
Classif.:  * 11N35 Sieves
                   11N05 Distribution of primes
Keywords:  small sieve; sifting by primes; coprime sifting set

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