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arXiv · 2609.29748

On smooth elements in arithmetic semigroups

Giovanna De Lauri·Dan Fretwell·Jack Ye
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Abstract

In this paper we explore the concept of smoothness in arithmetic semigroups. In particular, we give explicit asymptotics for the counting function $Ψ_G(x,y)$ when the arithmetic semigroup $G$ satisfies Axiom A (in the spirit of Knopfmacher). The asymptotics generalise the standard results in the literature for the integer case, and are expressed in terms of the Dickman $ρ$ function. Finally, we give applications of our results to counting smooth families of finite abelian groups and finite semisimple rings.

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Giovanna De Lauri, Dan Fretwell, Jack Ye. 2026-09-24. On smooth elements in arithmetic semigroups. https://arxiv.org/abs/2609.29748

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