TITLE: Why prove that multiplicative functions are a group with Dirichlet convolution?
QUESTION [9 upvotes]: Everyone likes to prove that Dirichlet convolution is a group operation on the multiplicative arithmetic functions, but what consequence does this have?
Does any important theorem use this fact?
Can general group theory lead to results about these functions (or even better, about numbers) from the this theorem?

Furthermore there are two ring structures on this set, the usual pointwise ring as well as the ring with convolution.
I would like to extend the same question for these.

REPLY [2 votes]: I saw a talk by John Thompson, not that I knew what it was about, looking at a copy of SL(2,Z) in Dirichlet series under convolution. This paper http://arxiv.org/abs/0803.1121 might be something...