Dirichlet Convolution is Commutative

Theorem
Let $f, g$ be arithmetic functions.

Let $*$ denote Dirichlet convolution.

Then:
 * $f * g = g * f$

Proof
From the definition of the Dirichlet convolution:


 * $\ds \map {\paren {f * g} } n = \sum_{a b \mathop = n} \map f a \map g b$

By definition, arithmetic functions are mappings from the natural numbers $\N$ to the complex numbers $\C$.

Thus $\map f a, \map g b \in \C$ and commutativity follows from Complex Multiplication is Commutative.

Also see

 * Properties of Dirichlet Convolution