E-mail. Jan.Snellman@math.su.se
Abstract.
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.