Entiers friables dans des progressions arithm\'etiques de grand module

TL;DR

This paper investigates the average error in counting y-friable integers in arithmetic progressions with large moduli, providing asymptotic results for thin sequences using specialized techniques.

Contribution

It introduces new methods to analyze the distribution of y-friable integers in arithmetic progressions with large moduli, extending previous results to thinner sequences.

Findings

Average error term asymptotic to -|a|Ψ(x/|a|, y)/2x

Valid in the range exp{(log log x)^{5/3+ε}} ≤ y ≤ x

Results apply on average over q ≤ x/M with M→∞

Abstract

We study the average error term in the usual approximation to the number of $y$-friable integers congruent to $a$ modulo $q$, where $a\neq 0$ is a fixed integer. We show that in the range $\exp\{(\log\log x)^{5/3+\varepsilon}\} \leq y \leq x$ and on average over $q\leq x/M$ with $M\rightarrow \infty$ of moderate size, this average error term is asymptotic to $-|a|\Psi(x/|a|,y)/2x$. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of $y$-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.