Möbius Inversion Formula/Abelian Group

Theorem

Let $G$ be an abelian group.

Let $f, g: \N \to G$ be mappings.

Then

$\ds \map f n = \prod_{d \mathop \divides n} \map g d$

if and only if:

$\ds \map g n = \prod_{d \mathop \divides n} \map f d^{\mu \paren {\frac n d} }$

where:

$d \divides n$ denotes that $d$ is a divisor of $n$

$\mu$ is the Möbius function.

Proof

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