Sieving the positive integers by large primes
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).
scientific article; zbMATH DE number 4042017
scientific article