Möbius Inversion Formula

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Theorem

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


Then:

$(1): \quad \displaystyle f \left({n}\right) = \sum_{d \mathop \backslash n} g \left({d}\right)$

iff:

$(2): \quad \displaystyle g \left({n}\right) = \sum_{d \mathop \backslash n} f \left({d}\right) \mu \left({\frac n d}\right)$

where:

$d \mathop \backslash n$ 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:

$\mu * u = \iota$


We have:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle f = g * u\) \(\implies\) \(\displaystyle \) \(\displaystyle f * \mu = \left({g * u}\right) * \mu\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\implies\) \(\displaystyle \) \(\displaystyle f * \mu = g * \left({u * \mu}\right)\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\implies\) \(\displaystyle \) \(\displaystyle f * \mu = g\) \(\displaystyle \) \(\displaystyle \)                    


Conversely:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle g = f * \mu\) \(\implies\) \(\displaystyle \) \(\displaystyle g * u = \left({f * \mu}\right) * u\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\implies\) \(\displaystyle \) \(\displaystyle g * u = f * \left({\mu * u}\right)\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\implies\) \(\displaystyle \) \(\displaystyle g * u = f\) \(\displaystyle \) \(\displaystyle \)                    

Hence the result.

$\blacksquare$


Source of Name

This entry was named for August Ferdinand Möbius.