Lois de r\'epartition des diviseurs des entiers friables

TL;DR

This paper demonstrates that the divisors of friable integers follow a Gaussian distribution under certain conditions, providing explicit error estimates and combining advanced probabilistic and analytic methods.

Contribution

It establishes the Gaussian law for divisors of friable integers with explicit error bounds and introduces new large deviations estimates for additive functions.

Findings

Divisors of friable integers follow a Gaussian distribution under standard conditions.

Explicit, near-optimal error estimates for the distribution are provided.

The approach combines saddle-point method with new large deviations estimates.

Abstract

According to a general probabilistic principle, the natural divisors of friable integers (i.e.~free of large prime factors) should normally present a Gaussian distribution. We show that this indeed is the case with conditional density tending to 1 as soon as the standard necessary conditions are met. Furthermore, we provide explicit, essentially optimal estimates for the decay of the involved error terms. The size of the exceptional set is sufficiently small to enable recovery of the average behaviour in the same optimal range. Our argument combines the saddle-point method with new large deviations estimates for the distribution of certain additive functions.