Definition:Dirichlet Convolution

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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.