# Definition:Dirichlet Convolution

## Definition

Let $f, g$ be arithmetic functions.

### Definition 1

The Dirichlet convolution of $f$ and $g$ is the arithmetic function:

$\ds \map {\paren {f * g} } n = \sum_{d \mathop \divides n} \map f d \map g {\frac n d}$

where the summation runs over the set of positive divisors $d$ of $n$.

### Definition 2

The Dirichlet convolution of $f$ and $g$ is the arithmetic function:

$\ds \map {\paren {f * g} } n = \sum_{a b \mathop = n} \map f a \map g b$

where the summation runs over all pairs of positive integers $\tuple {a, b}$ with $a b = n$.

## Also see

• Results about Dirichlet convolution can be found here.

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.