Let $S$ be a non-trivial subsemiring of the ring of integers of a totally real number field, and denote by $D_S$ the integral semidomain of all arithmetic functions $\mathbf N^+ \to S$ with the operations of pointwise addition and Dirichlet convolution. It is clear that $D_S$ is atomic (respectively, factorial) only if so is $S$, and my questions are somehow on the converse of this statement:
Q1. Is there any case at all in which $D_S$ is atomic (respectively, factorial)? Is there anything in the literature about that?
I'm especially interested in the case when $S$ is a ring, and don't even know the answer for $S = \mathbf Z$.
Let me recall that an integral semidomain is atomic if every non-zero, non-unit element is a (finite) product of atoms (irreducible elements) in at least one way, and factorial if every non-zero, non-unit element can be expressed as a product of atoms in an essentially unique way.
Q2. What about the atomicity of the subrig of $D_S$ consisting of finitely supported arithmetic functions?