Definition:Dirichlet Convolution

Definition
Let $f, g$ be arithmetic functions.

The Dirichlet convolution of $f$ and $g$ is defined to be:


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

where $d \mathop \backslash n$ denotes that $d$ is a divisor of $n$.

This is trivially equivalent to:


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

Also see

 * Properties of Dirichlet Convolution