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


Jan Snellman

Department of Mathematics, Stockholm University, SE-10691 Stockholm, Sweden


 We study $(\UNI,+,\oplus)$, the ring of arithmetical  functions with unitary  convolution,  giving an isomorphismbetween $({\mathcal A},+,\oplus)$ and  a generalized power series  ring on infinitely many variables, similar to theisomorphism of Cashwell-Everett \cite{NumThe}  between the ring $({\mathcal A},+,\cdot)$ of arithmetical functions with  Dirichlet convolution  and the power series ring ${\mathbb C} [\![x_1,x_2,x_3,\dots]\!]$ on  countably  many variables. We topologize it with respect to a natural norm, and  show 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.

AMSclassification. 11A25, 13J05, 13F25

Keywords.  Unitary convolution, Schauder Basis, factorization into atoms, zero divisors.