Une Remarque sur une Formule Sommatoire liée à la Somme des Chiffres

Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that ?_{n ￡ x} f(n)=o(x)\sum_{n\le x} f(n)=o(x) . We propose sufficient conditions on the function f to deduce the equality ?_{n ￡ x} f(s(n))=o(x)\sum_{n\le x} f(s(n))=o(x) . Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))^α, where α is a positive real number.