PreprintCC BY 4.0arXiv
On smooth elements in arithmetic semigroups
Abstract
In this paper we explore the concept of smoothness in arithmetic semigroups. In particular, we give explicit asymptotics for the counting function ΨG(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.
Related
- Arithmetic growth in partially commutative semigroups of endomorphisms
- Smooth numbers in simultaneous arithmetic progressions
- Arithmetic progressions in dense subsets of primitive elements of finite fields
- A new hierarchy for automaton semigroups
- Kobayashi's problem for power semigroups of inverse semigroups
BibTeX
@misc{lauri2026smooth,
title = {{On smooth elements in arithmetic semigroups}},
author = {Giovanna De Lauri and Dan Fretwell and Jack Ye},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29748},
eprint = {2609.29748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29748},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On smooth elements in arithmetic semigroups
AU - Giovanna De Lauri
AU - Dan Fretwell
AU - Jack Ye
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29748
UR - https://arxiv.org/abs/2609.29748
AB - 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.
ER -