Répartition sur les entiers friables des valeurs de fonctions arithmétiques liées aux fonctions de Piltz

Abstract. An integer greater than one is said to be \(y\)-friable, where \(y\) is a given real number, if all its prime factors do not exceed \(y\). In this paper, we establish an asymptotic formula for the mean value, on \(y\)-friable integers, of functions which belongs to a large class of multiplicative arithmetical functions, namely those functions whose Dirichlet series behaves, in the neighborhood of 1, essentially as a power of the Riemann Zeta Function. As applications, we obtain several estimates for the mean values of arithmetical functions, in particular over \(y\)-friable integers in arithmetical progressions. G. Hanrot, G. Tenenbaum and J. Wu studied t he same problem for a wide class of arithmetical functions which contains a subclass we study here. However, they used a global method which confines their results in relatively short ranges. We use here a local method which improves significantly the ranges of validity of known results.