Purpose

The Möbius inversion formula allows recovery of an arithmetic function $g$ from its summatory version $f$ defined by $f\left(n\right)={\sum }_{\mathrm{d|n}}^{}g\left(d\right)$. Given $f$ for $n$ up to some limit, the inversion states $g\left(n\right)={\sum }_{\mathrm{d|n}}^{}\mu \left(\frac{n}{d}\right)f\left(d\right)$ where $\mu$ is the Möbius function.

The Möbius function is defined as $\mu \left(n\right)=\left\{\begin{array}{cc}1& \text{if}n=1\\ 0& \text{if}n\text{has a squared prime factor}\\ \mathrm{-1}& \text{if}n\text{is squarefree with an odd number of prime factors}\end{array}\right\}$. Its multiplicative property makes Möbius inversion a key result in multiplicative number theory.