On the small sieve. I. Sifting by primes
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.
scientific article; zbMATH DE number 3677949
scientific article