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:


 * $\displaystyle \left({f * g}\right) \left({n}\right) = \sum_{a b \mathop = n} f \left({a}\right) g \left({b}\right)$

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

Thus $f \left({a}\right), g \left({b}\right) \in \C$ and commutativity follows from commutativity of multiplication of complex numbers.

Also see

 * Properties of Dirichlet Convolution