Entiers friables dans des progressions arithmétiques de grand module

Abstract

We study the average error term in the usual approximation to the number of $y$-friable integers congruent to $a$ modulo $q$, where $a\ne 0$ is a fixed integer. We show that in the range $\exp \{(\log \log x{)}^{5/3+\epsilon }\}\le y\le x$ and on average over $q\le x/M$ with $M\to \infty$ 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.

29 pages, in French