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

TL;DR

This paper derives rapidly convergent formulas for the distribution of smooth numbers, including special cases like 3-smooth, 5-smooth, and 7-smooth, extending classical results by Hardy and Littlewood.

Contribution

It introduces new, rapidly convergent formulas for the distribution of smooth numbers, generalizing and refining previous classical formulas.

Findings

Derived formulas for 3-smooth, 5-smooth, 7-smooth numbers

Extended Hardy and Littlewood's formula for $N_{a,b}(x)$

Provided proofs for the convergence and accuracy of these formulas

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 $x$.