Question: This question was originally posted in [Elements of finite order in the group of arithmetic functions under Dirichlet convolution.](https://math.stackexchange.com/questions/265446/elements-of-finite-order-in-the-group-of-arithmetic-functions-under-dirichlet-co/266702#comment582643_266702)

and it goes as follows: Let G be the group consisting of all arithmetic functions (i.e. functions $f:\mathbb{N} \to \mathbb{C}$) under the "convolution"operation $\ast,$ defined as $$(f\ast g)(n):=\sum\_{ab=n}f(a)g(b), n \in \mathbb{N}.$$
(Note that each function in the group assigns the value $1$ to $1.$)

Alexander Gruber showed in the aforementioned post that G is a torsion-free abelian group. A few further facts can be easily proved:

(1) G is not finitely generated and in particular is infinite.

(2)If one lets H to denote the subgroup of G consisted of multiplicative functions, then $|G:H|=\infty.$

However my question is: What is known in the literature about the group structure of G ?

Answer: 
$G$ is a certain subgroup of the multiplicative group of the *ring* of all arithmetic functions under convolution and pointwise addition. This ring is isomorphic to the ring of formal [Dirichlet series](http://en.wikipedia.org/wiki/Dirichlet_series), which is in turn isomorphic to the ring of formal power series $\mathbb{C}[[x\_1, x\_2, ... ]]$ in countably many variables, one for each prime (write out $\frac{1}{n^s}$ in terms of the prime factorization of $n$). 

So the question reduces to the following: what is the structure of the group of complex power series in countably many variables with constant term $1$ under multiplication?

To answer that question, observe that the logarithm of such a formal power series is well-defined using the formal power series of $\log (1 + x)$, and the result is a formal power series with constant term $0$, and conversely. Hence the group in question is isomorphic to the additive group of formal power series in countably many variables with constant term $0$, which is just a countable product of copies of $\mathbb{C}$, or in purely group-theoretic terms a vector space over $\mathbb{Q}$ of dimension the cardinality of $\mathbb{R}$.
"However my question is: What is known in the literature about the group structure of G ?"

JOURNAL ARTICLE

Groups of Arithmetical Functions

James E. Delany

Mathematics Magazine

Vol. 78, No. 2 (Apr., 2005), pp. 83-97