Propri\'et\'es multiplicatives des entiers friables translat\'es

TL;DR

This paper investigates the multiplicative properties of shifted friable integers, extending previous results, and demonstrates a phase transition in their behavior, including an Erd"os-Kac-type theorem, using recent distributional work.

Contribution

It extends the understanding of multiplicative functions over shifted friable integers and establishes a phase transition in their statistical behavior, including an improved Erd"os-Kac theorem.

Findings

Estimates mean values of arithmetic functions over shifted friable numbers.

Shows a change in behavior depending on the ratio of f4g y to fclog x.

Proves an Erdf6s-Kac-type theorem for shifted friable numbers.

Abstract

An integer is said to be $y$-friable if its greatest prime factor P(n) is less than $y$. In this paper, we study numbers of the shape $n-1$ when $P(n)\leq y$ and $n\leq x$. One expects that, statistically, their multiplicative behaviour resembles that of all integers less than $x$. Extending a result of Basquin, we estimate the mean value over shifted friable numbers of certain arithmetic functions when $(\log x)^c \leq y$ for some positive $c$, showing a change in behaviour according to whether $\log y / \log\log x$ tends to infinity or not. In the same range in $(x, y)$, we prove an Erd\"os-Kac-type theorem for shifted friable numbers, improving a result of Fouvry and Tenenbaum. The results presented here are obtained using recent work of Harper on the statistical distribution of friable numbers in arithmetic progressions.