Dirichlet Convolution is Associative

Theorem
Let $f, g, h$ be arithmetic functions.

Let $*$ denote Dirichlet convolution.

Then:
 * $\paren {f * g} * h = f * \paren {g * h}$

Proof
We have:

and

and associativity follows.

Also see

 * Properties of Dirichlet Convolution