Dirichlet Convolution is Associative

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

Let $*$ denote Dirichlet convolution.

Then $\left({f * g}\right) * h = f * \left({g * h}\right)$

Proof
We have:

and

and associativity follows.

Also see

 * Properties of Dirichlet Convolution