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**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,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 **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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