On the average distribution of divisors of friable numbers

TL;DR

This paper establishes a central limit theorem for the distribution of divisors of y-friable numbers less than x, extending previous results to a broader range of y using a two-variable saddle-point method.

Contribution

It proves a central limit theorem on average for divisors of y-friable numbers over a wider range of y, improving upon prior constraints.

Findings

Extended the range of y for which the CLT holds

Applied a two-variable saddle-point method for the proof

Demonstrated the distribution of divisors follows a normal distribution on average

Abstract

A number is said to be $y$-friable if it has no prime factor greater than $y$. In this paper, we prove a central limit theorem on average for the distribution of divisors of $y$-friable numbers less than $x$, for all $(x, y)$ satisfying $2\leq y \leq {\rm e}^{(\log x)/(\log\log x)^{1+\varepsilon}}$. This was previously known under the additional constraint $y\geq {\rm e}^{(\log\log x)^{5/3+\varepsilon}}$, by work of Basquin. Our argument relies on the two-variable saddle-point method.