Average of some multiplicative functions on the set of integers without large prime factor

Abstract

Let λ > 0 , δ ∈]0, 1[ and f (n) be a multiplicative function satisfying essentially for a large x ; f (p)≤x log f (p) = λx+ O(x(log x) −δ). In the present work, we establish an asymptotic formula for the two sums f (n)≤x;P (n)≤y 1 f (n) and f (n)≤x;P (n)≤y 1 , valid in the domain 1 ≤ cx ≤ 1 C 0 (log f (y)) δ , for a suitable constants c and C 0 .