Definition:Dirichlet Convolution

Definition
Let $f, g$ be arithmetic functions.

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


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

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

This is trivially equivalent to:


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

Also see

 * Properties of Dirichlet Convolution