Fonctions moyennes sur les entiers sans facteurs carrés et $y$-friables

Abstract. A natural number greater than 1 is said to be $y$-smooth, $y$ real, if all its prime factors do not exceed $y$. In this work, we are interested in the estimates of means on integers free of Prime factors ${}^{>}y$ of the function $f_g(n)=\frac{1}{h(n)}\sum _{d/n}g(d)f(d)$, where $h=g\ast 1$, $f$ and $g$ belong to a class $S$ containing the set $\{\frac{\sigma (n)}{n},\frac{n}{\phi (n)},\frac{\phi (n,l)}{n}\}$, where function $S_{fg}(x,y)=\sum _{n\le x,P(n)\le y}{f}_g(n)$ is proportional to the summation function of integers without square factors and $y$-smooth. This study generalizes that by J.-M. De Koninck and J. Grah for the function $f_{{\mu }^2}$ and that of M. Naîmi for the mean over the friable integers of the function ${\mu }^{2}f$.