算术半群中的光滑元素

中文摘要 AI 辅助

本文探讨了算术半群中光滑性的概念。特别地，当算术半群$G$满足公理A（遵循Knopfmacher的思想）时，我们给出了计数函数$\u0053Psi_G(x,y)$的显式渐近式。这些渐近式推广了文献中关于整数情形的标准结果，并以Dickman $\u0053rho$函数表示。最后，我们将结果应用于计数有限阿贝尔群和有限半单环的光滑族。

英文摘要

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.

发表机构

• Lancaster University（兰卡斯特大学）

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