On the average distribution of divisors of friable numbers

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.