Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: De Lauri, Giovanna, Fretwell, Dan, Ye, Jack
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214392556