Number of prime factors with a given multiplicity

Abstract

Let k ⩾ 1 be a natural number and ωk (n) denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions ωk with k ⩾ 1. Moreover, we prove that the function ω1(n) has normal order log log n and the function (ω1(n) − log log n)/√log log n has a normal distribution. Finally, we prove that the functions ωk (n) with k ⩾ 2 do not have normal order F(n) for any nondecreasing nonnegative function F.