Properties of Dirichlet Convolution

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

Let $*$ denote Dirichlet convolution.

Let $\iota$ be the identity arithmetic function.

Then:

Dirichlet Convolution is Commutative
$f * g = g * f$

Dirichlet Convolution is Associative
$\left({f * g}\right) * h = f * \left({g * h}\right)$

Identity Element for Dirichlet Convolution
$\iota * f = f$

Dirichlet Convolution Preserves Multiplicativity
If $f$ and $g$ are multiplicative, then $f*g$ is multiplicative.

Also see

 * Definition:Ring of Arithmetic Functions