Definition:Dirichlet Convolution
Jump to navigation
Jump to search
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
- Results about Dirichlet convolution can be found here.
Generalization
Source of Name
This entry was named for Johann Peter Gustav Lejeune Dirichlet.