On the average distribution of divisors of friable numbers

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 $2\le y\le {\mathrm{e}}^{(\log x)/(\log \log x{)}^{1+\epsilon }}$. This was previously known under the additional constraint $y\ge {\mathrm{e}}^{(\log \log x{)}^{5/3+\epsilon }}$, by work of Basquin. Our argument relies on the two-variable saddle-point method.

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• Lois de répartition des diviseurs des entiers friables