Propriétés multiplicatives des entiers friables translatés

Sary Drappeau¹

¹ Centre de recherches mathématiques Université de Montréal CP 6128, succ. Centre-Ville Montréal H3C 3J7, Canada

Colloquium Mathematicum, Tome 137 (2014) no. 2, pp. 149-164.

Résumé

An integer $n$ 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 (2010), 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 $\log y /\log x$ tends to infinity or not. In the same range in $(x, y)$, we prove an Erdős–Kac-type theorem for shifted friable numbers, improving a result of Fouvry Tenenbaum (1996). The results presented here are obtained using recent work of Harper (2012) on the statistical distribution of friable numbers in arithmetic progressions.

Détail

DOI : 10.4064/cm137-2-1

Mots-clés : integer said y friable its greatest prime factor paper study numbers shape n leq leq expects statistically their multiplicative behaviour resembles integers extending result basquin estimate mean value shifted friable numbers certain arithmetic functions log leq positive showing change behaviour according whether log log tends infinity range prove erd kac type theorem shifted friable numbers improving result fouvry amp tenenbaum results presented here obtained using recent work harper statistical distribution friable numbers arithmetic progressions