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The Möbius inversion formula is a useful tool allowing to calculate sums of arithmetic functions. It was developed by August Ferdinand Möbius (1790 – 1868).

## Theorem: Möbius Inversion Formula

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 $\alpha:$

$$\alpha(n)=\sum_{d\mid n}\mu(d)\beta\left(\frac nd\right).$$

| | | | | created: 2019-04-06 22:24:59 | modified: 2019-04-06 22:42:08 | by: bookofproofs | references: [701], [1272]

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