The average prime divisor of an integer in short intervals

Abstract

and as usual co(n) is the number of distinct prime factors of n, f2(n) is the total number of prime factors of n, f II n means that p~ (p prime) exactly divides n. It was shown in [1] that, for each fixed natural number m, there exist computable constants el, e z, ..., era, fa,f2 ..... fro, 0 2). These questions were considered in a general setting by De Koninck and Mercier [2], where under suitable conditions asymptotic relations of the form (1.3) Y~ f(P(n)) = (1 + o(1)) Z f(n) (x ~co)