A property of the counting function of integers with no large prime factors
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.)
scientific article; zbMATH DE number 3918483