Th\'eor\`emes de type Fouvry--Iwaniec pour les entiers friables

TL;DR

This paper establishes a significant distribution property of y-friable integers less than x, demonstrating they are well distributed in residue classes on average, with applications to estimating sums involving the divisor function.

Contribution

It proves a weak exponent of distribution for y-friable integers in a new range and improves previous results by combining advanced methods and recent work on friable integers.

Findings

y-friable integers have a distribution exponent at least 3/5 - ε

Improved estimates for sums involving τ(n-1) over friable integers

Enhanced understanding of the distribution of friable integers in arithmetic progressions

Abstract

An integer n is said to be y-friable if its largest prime factor $P^+(n)$ is less than y. In this paper, it is shown that the y-friable integers less than x have a weak exponent of distribution at least $3/5-\varepsilon$ when $(\log x)^c\leq y\leq x^{1/c}$ for some $c=c(\varepsilon)\geq 1$, that is, they are well distributed in the residue classes of a fixed integer $a$, on average over moduli $\leq x^{3/5-\varepsilon}$ for each fixed $a\neq 0$ and $\varepsilon>0$. We present an application to the estimation of the sum $\sum_{2\leq n\leq x, P^+(n)\leq y}\tau(n-1)$ when $(\log x)^c\leq y$. This follows and improves on previous work of Fouvry and Tenenbaum. Our proof combines the dispersion method of Linnik in the setting of Bombieri, Fouvry, Friedlander and Iwaniec, and recent work of Harper on friable integers in arithmetic progressions.