On Integers Free of Large Prime Factors

The number <!-- MATH $\Psi (x,y)$ --> of integers and free of prime factors y$"> has been given satisfactory estimates in the regions <!-- MATH $y \leq {(\log x)^{3/4 - \varepsilon }}$ --> and <!-- MATH $y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\}$ --> \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 <!-- MATH $\Psi (x,y)$ --> uniformly for <!-- MATH $x \geq y \geq 2$ --> within a factor <!-- MATH $1 + O((\log y)/(\log x) + (\log y)/y)$ --> . As an application, we derive a simple formula for <!-- MATH $\Psi (cx,y)/\Psi (x,y)$ --> , where <!-- MATH $1 \leq c \leq y$ --> . We also prove a short interval estimate for <!-- MATH $\Psi (x,y)$ --> .