Möbius Inversion Formula

Möbius Inversion Formula (by August Ferdinand Möbius) states that if ` Df ` the summatory function of f

then

 

where ` f ` is an arithmetic function.  This is more like a "formula machine", which, when given an input ` f `, gives you a version of formula as an output (Well, as long as you know what ` Df ` is).  ` \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 ` 1(n) ` function is the Möbius function ` \mu(n) `

the formula can be proven succinctly as follows.
Proof:     ` \mu \mbox{*} Df` `
` =  \mu \mbox{*} (1 \mbox{*} f) =  (\mu \mbox{*} 1) \mbox{*} f   =  delta_1 \mbox{*} f    =    f  `
(proven)

Example
For `f(n) = \phi(n)` the Euler totient function, we know that

Hence applying Möbius Inversion we have

i.e.                                     

since ` \phi(n) = (\mu * D\phi)(n)  = \sum_{d|n}\mu(d) \cdot D\phi(\frac{n}{d})  = \sum_{d|n}\mu(d) \cdot I(\frac{n}{d}) = \sum_{d|n}\mu(d) \cdot \frac{n}{d} `