Questions tagged [mobius-inversion]

In number theory, the Möbius inversion formula states that if $f$ and $g$ are arithmetic functions for which

$$g(n) = \sum_{d | n} f(d)$$

for every integer $n \ge 1$, then we can recover $f$ by

$$f(n) = \sum_{d | n} \mu(d) g\left(\frac{n}{d}\right)$$

where $\mu$ is the Möbius function. Written in terms of Dirichlet convolutions,

$$g = f \ast 1 \implies f = \mu \ast g$$

Reference: Möbius inversion formula.