On the distribution of the total number of generators of $h$-free and $h$-full elements in an abelian monoid

Abstract

Let $\mathfrak{m}$ be an element of an abelian monoid, with $\Omega(\mathfrak{m})$ denoting the total number of prime elements generating $\mathfrak{m}$. We study the moments of $\Omega(\mathfrak{m})$ over subsets of $h$-free and $h$-full elements, establishing the normal order of $\Omega(\mathfrak{m})$ within these subsets. This work continues the study on the distribution of generalized arithmetic functions over $h$-free and $h$-full elements in abelian monoids as introduced in the authors' previous work.