Properties of Dirichlet Convolution

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

Let $*$ denote the Dirichlet convolution of two arithmetic functions.

Let $\iota$ be the identity arithmetic function.

Then:


 * $(1): \quad f * g = g * f$


 * $(2): \quad (f * g) * h = f * (g * h)$


 * $(3): \quad \iota * f = f$

Proof
$(1): \quad$ From the definition of the Dirichlet convolution:


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

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.

$(2): \quad$ We have

and

and associativity follows.

$(3): \quad$ We have:

and we are done.