I found this article of E.D. Cashwell and C.J. Everett "The ring of number-theoretic functions" and they said Dirichlet series ring is isomorphic to formal power series ring of countably many variables.
We can view the formal Dirichlet series as the ring from the article.
However, they didn't prove this and I have no idea where to even start.
We need to bijectively send
$$\sum_{n\geq 1}a_n n^{-s}$$ to $$\sum_{n1, n2,\ldots \in \mathbb N} b_nx_{i_1}^{n_{i_1}}\ldots x_{i_k}^{n_{i_k}}$$
I was thinking of splitting the $n$ from Dirichlet series into prime factors and somehow send parts of $a_n$, but I don't know if that even makes sense.