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Keywords:
omega function; Abelian monoid; the first moment; the second moment; $h$-free element; $h$-full element
omega function; Abelian monoid; the first moment; the second moment; $h$-free element; $h$-full element
Summary:
Let ${\mathfrak m}$ be an element of an Abelian monoid, and let $\Omega ({\mathfrak m})$ denote the total number of prime elements (counted with multiplicity) generating ${\mathfrak m}$. We investigate the distribution of $\Omega ({\mathfrak m})$ over the subsets of $h$-free and $h$-full elements, obtaining moment estimates and establishing its normal order within these subsets. This extends the authors' previous work (see S. Das et al., 2025c) on $\omega ({\mathfrak m})$, where multiplicities of prime elements were not considered. In particular, we develop new identities involving sums over prime elements, which play a central role in the analysis. Several applications are presented, including ideals in number fields, effective divisors in global function fields, and effective zero-cycles on geometrically irreducible projective varieties.
References:
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