A property of the counting function of integers with no large prime factors
type: Item
instance of: scholarly article
title: A property of the counting function of integers with no large prime factors
zbMATH Open document ID: 0575.10039
DOI: 10.1016/0022-314X(86)90030-2
author: Douglas Hensley
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Psi\) (x,y) denote the number of positive integers \(\leq x\), all of whose prime factors are \(\leq y\). Approximations to \(\Psi\) (x,y) by smooth functions have been given by various authors for successively larger ranges, the current record being the range \(\log y\geq (\log \log x)^{5/3+\epsilon}.\) In the paper under review, the author considers for fixed c the quantity \(\Psi\) (cx,y)/\(\Psi\) (x,y) and shows that it behaves regularly in a much larger range. His main result is that the relation \(\Psi\) (cx,y)\(\sim c \Psi (x,y)\) holds uniformly for \(\epsilon\leq c\leq 1\) \((\epsilon >0\) fixed), provided \(x\to \infty\) and log y/log log \(x\to \infty.\) (Remark: For more precise estimates of \(\Psi\) (cx,y)/\(\Psi\) (x,y) see a forthcoming paper by \textit{G. Tenenbaum} and the reviewer in the Trans. Am. Math. Soc.)
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 3918483
zbMATH Keywords: integers free of large prime factors
zbMATH Keywords: approximations by smooth functions
reviewed by: Adolf Hildebrand
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3918483