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

TL;DR

This paper investigates the distribution of the number of distinct prime generators of elements in abelian monoids, focusing on h-free and h-full elements, and applies the results to various mathematical structures.

Contribution

It introduces a unified framework for analyzing the distribution of prime generators in h-free and h-full elements within abelian monoids, extending to number fields and algebraic varieties.

Findings

Established the normal order of the number of prime generators in these elements

Derived moments of the prime generator count over specific subsets

Applied results to number fields, function fields, and algebraic varieties

Abstract

This work introduces the first in-depth study of h-free and h-full elements in abelian monoids, providing a unified approach for understanding their role in various mathematical structures. Let m be an element of an abelian monoid, with {\omega}(m) denoting the number of distinct prime elements generating m. We study the moments of {\omega}(m) over subsets of h-free and h-full elements, establishing the normal order of {\omega}(m) within these subsets. Our findings are then applied to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of this approach.