Möbius Inversion Formula

Theorem
Let $f$ and $g$ be arithmetic functions.

Then:


 * $(1): \quad \ds \map f n = \sum_{d \mathop \divides n} \map g d$




 * $(2): \quad \ds \map g n = \sum_{d \mathop \divides n} \map f d \, \map \mu {\frac n d}$

where:
 * $d \divides n$ denotes that $d$ is a divisor of $n$
 * $\mu$ is the Möbius function.

Proof
Let $u$ be the unit arithmetic function and $\iota$ the identity arithmetic function.

Let $*$ denote Dirichlet convolution.

Then equation $(1)$ states that $f = g * u$ and $(2)$ states that $g = f * \mu$.

The proof rests on the following facts:

By the lemma to Sum of Möbius Function over Divisors:
 * $\mu * u = \iota$

By Properties of Dirichlet Convolution, Dirichlet convolution is commutative, associative and $h * \iota = h$ for all $h$.

We have:

Conversely:

Hence the result.