Moyenne de la fonction de Piltz sur les entiers sans facteur premier dans un intervalle

Abstract. In this work, we study the average behavior of the Piltz function "\(\tau_k\)", where \(k\) is a strictly positive real number, on integers without prime factors in a given interval. This study completes the work of Andreas Weingartner and Gérald Tenenbaum concerning the function \(1(n)=1\). From this work, we deduce the estimate of the function \(\tau_k\) on integers having at most two prime factors in an interval \(\bigl]x^{1/3},x^{1/2}\bigr]\), where \(x\) is a real number \(\geqslant 2\).