On Integers Free of Large Prime Factors

Abstract

The number $\Psi (x,y)$ of integers $\leq x$ and free of prime factors $> y$ has been given satisfactory estimates in the regions $y \leq {(\log x)^{3/4 - \varepsilon }}$ and $y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\}$. In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates $\Psi (x,y)$ uniformly for $x \geq y \geq 2$ within a factor $1 + O((\log y)/(\log x) + (\log y)/y)$. As an application, we derive a simple formula for $\Psi (cx,y)/\Psi (x,y)$, where $1 \leq c \leq y$. We also prove a short interval estimate for $\Psi (x,y)$.