The Möbius inversion formula is a useful tool allowing to calculate sums of arithmetic functions. It was developed by August Möbius (1790 – 1868).
Let $\alpha:\mathbb N\to \mathbb C$ be an arbitrary arithmetic function and let $\beta:\mathbb N\to\mathbb C$ be another arithmetic function given by $$\beta(n):=\sum_{d\mid n}\alpha(d).$$ Then, using the Möbius function, we can reverse the equation and provide a formula for $\alpha:$
$$\alpha(n)=\sum_{d\mid n}\mu(d)\beta\left(\frac nd\right).$$
Proofs: 1
