The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers

Abstract

In this paper we present and prove rapidly convergent formulas for the distribution of the $3$-smooth, $5$-smooth, $7$-smooth and all other smooth numbers. One of these formulas is another version of a formula due to Hardy and Littlewood for the arithmetic function $N_{a,b}(x)$, which counts the number of positive integers of the form $a^{p}b^{q}$ less than or equal to $x$.