Möbius Inversion Formula

Möbius Inversion Formula (by August Ferdinand Möbius) states that if ${\displaystyle Df}$ the summatory function of f

then

 

where ${\displaystyle f}$ is an arithmetic function.  This is more like a "formula machine", which, when given an input ${\displaystyle f}$, gives you a version of formula as an output (Well, as long as you know what ${\displaystyle Df}$ is).  ${\displaystyle \mu }$ is the Möbius function.  [ By the way, Möbius was also famous for his Möbius strip. ]  Using Dirichlet Convolutions (* operation), the formula just says that

and since the *-inverse of the ${\displaystyle 1\left(n\right)}$ function is the Möbius function ${\displaystyle \mu \left(n\right)}$

the formula can be proven succinctly as follows.
Proof:     ${\displaystyle \mu \text{*}Df}$
${\displaystyle = \mu \text{*}\left(1\text{*}f\right)= \left(\mu \text{*}1\right)\text{*}f = {\delta }_1\text{*}f = f }$
(proven)

Example
For ${\displaystyle f\left(n\right)=\phi \left(n\right)}$ the Euler totient function, we know that

Hence applying Möbius Inversion we have

i.e.                                     

since ${\displaystyle \phi \left(n\right)=(\mu \cdot D\phi )\left(n\right) =\sum _{d|n}\mu \left(d\right)\cdot D\phi \left(\frac{n}{d}\right) =\sum _{d|n}\mu \left(d\right)\cdot I\left(\frac{n}{d}\right)=\sum _{d|n}\mu \left(d\right)\cdot \frac{n}{d}}$