Identity Element for Dirichlet Convolution

Theorem

Let $f$ be an arithmetic function.

Let $*$ denote Dirichlet convolution.

Let $\iota$ be the identity arithmetic function.

Then:

$\iota * f = f$

Proof

We have:

\(\ds \map {\paren {\iota * f} } n\)

\(=\)

\(\ds \sum_{d \mathop \divides n} \delta_{d 1} \map f {\frac n d}\)

\(\ds \)

\(=\)

\(\ds \map f n\)

Hence the result.

$\blacksquare$

Also see

• Properties of Dirichlet Convolution