Dirichlet Convolution is Associative

Theorem

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

Let $*$ denote Dirichlet convolution.

Then:

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

Proof

We have:

\(\ds \map {\paren {\paren {f * g} * h} } n\)

\(=\)

\(\ds \sum_{a b \mathop = n} \map {\paren {f * g} } a \map h b\)

\(\ds \)

\(=\)

\(\ds \sum_{a b \mathop = n} \ \sum_{c d \mathop = a} \map f c \map g d \map h b\)

\(\ds \)

\(=\)

\(\ds \sum_{b c d \mathop = n} \map f c \map g d \map h b\)

and

\(\ds \map {\paren {f * \paren {g * h} } } n\)

\(=\)

\(\ds \sum_{a b \mathop = n} \map f a \map {\paren {g * h} } b\)

\(\ds \)

\(=\)

\(\ds \sum_{a b \mathop = n} \map f a \sum_{c d \mathop = b} \map g c \map h d\)

\(\ds \)

\(=\)

\(\ds \sum_{a c d \mathop = n} \map f a \map g c \map h d\)

and associativity follows.

$\blacksquare$

Also see

• Properties of Dirichlet Convolution