The Möbius inversion formula is a useful tool allowing to calculate sums of arithmetic functions. It was developed by August 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 $\alpha:$

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

Proofs: 1

Proofs: 1
Sections: 2 3

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  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
  2. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927