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This question was originally posted in Elements of finite order in the group of arithmetic functions under Dirichlet convolution.

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 ?

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    $f(1) = 1$ isn't part of the definition of an arithmetic function; please include this condition explicitly in the definition of $G$ (or else it isn't a group).2012-12-29

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