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.

$\blacksquare$

Also see

• Properties of Dirichlet Convolution