Mobius inversion formula
The Möbius Inversion Formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. Originally proposed by August Ferdinand Möbius in 1832, it has many uses in Number Theory and Combinatorics.
Let and be arithmetic functions and denote the Möbius Function. Then it follows that
: Notice the double implication, so we have two directions to prove. We proceed with the proof of the backwards direction first. We have
To finish, we will use the fact that
If we have then we have
and that if otherwise. Hence by considering we get
The first direction is satisfied, and now we must prove the second. We see that
Both directions have been proven, which completes our work
One of the most common applications of the formula is by proving that
While there are some common combinatorial and group theoretic arguments one could use, a Möbius Inversion Formula solution also suffices. Clearly by choosing and the theorem is proven.