Definition:Dirichlet Convolution

Definition

Let $f, g$ be arithmetic functions.

Definition 1

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

$\displaystyle \left({f * g}\right) \left({n}\right) := \sum_{d \mathop \backslash n} f \left({d}\right) g \left({\frac n d}\right)$

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:

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

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

Also see

• Equivalence of Definitions of Dirichlet Convolution
• Properties of Dirichlet Convolution

• Results about Dirichlet convolution can be found here.

Generalization

• Definition:Convolution of Mappings on Divisor-Finite Monoid

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.