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 \(f_g(n)=\frac{1}{h(n)}\sum\limits_{d/n}g(d)f( d)\), where \(h=g*1\), \(f\) and \(g\) belong to a class \(S\) containing the set \(\{\frac{\sigma(n)}{n},\frac{ n}{\varphi(n)},\frac{\varphi(n,l)}{n}\}\), 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 \(\mu^2f\).