Properties of Dirichlet Convolution

Theorem

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

Let $*$ denote Dirichlet convolution.

Let $\iota$ be the identity arithmetic function.

Then the following properties hold:

Dirichlet Convolution is Commutative

$f * g = g * f$

Dirichlet Convolution is Associative

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

Identity Element for Dirichlet Convolution

$\iota * f = f$

Dirichlet Convolution Preserves Multiplicativity

Let $f, g: \N \to \C$ be multiplicative arithmetic functions.

Then their Dirichlet convolution $f * g$ is again multiplicative.

Also see

• Definition:Ring of Arithmetic Functions