The ring of arithmetical functions with unitary convolution: Divisorial and topological properties

TL;DR

This paper explores the algebraic and topological structure of the ring of arithmetical functions under unitary convolution, establishing isomorphisms, ideal properties, and factorization characteristics.

Contribution

It introduces a new isomorphism to a generalized power series ring and analyzes the ideal and factorization properties of the ring under a natural topology.

Findings

Isomorphism to a generalized power series ring on infinitely many variables

All ideals are quasi-finite under the natural norm topology

Existence of many non-associate regular non-units

Abstract

We study the ring of arithmetical functions with unitary convolution, giving an isomorphism to a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett between the ring of arithmetical functions with Dirichlet convolution and the power series ring on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units.