On the algebraic properties of the ring of Dirichlet convolutions

Abstract

The most important case, largely studied in analytic number theory, is the case when R is a domain (or even more particullary, when R = C) and Γ = N∗ is the multiplicative monoid of positive integers. Cashwell and Everett showed that F(N∗, R) is also a domain. Moreover, if R is an UFD with the property that R[[x1, ... , xn]] are UFD for any n ≥ 1, then F(N∗, R) is also an UFD.