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

TL;DR

This paper investigates the distribution and moments of the total number of prime generators of elements in an abelian monoid, focusing on $h$-free and $h$-full subsets, and establishes their normal order.

Contribution

It extends previous work by analyzing the moments and normal order of the prime generator count in $h$-free and $h$-full elements within abelian monoids.

Findings

Established the normal order of the total number of prime generators.

Derived the moments of the distribution of prime generators.

Extended the distribution analysis to $h$-free and $h$-full elements.

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.