To derive this from Möbius inversion formula as you ask, remember that where
So in fact where (what is this sum equal to? apply Möbius inversion formula )
Remark: Although this works, the problem is that Möbius inversion formula is first derived from the value of to start with ( another way would be to derive the above result from the fact that is multiplicative -factorizing the sum-, and then prove the inversion).
Let (to avoid writing it out everywhere)
By (2) we have :
Here comes the trick:
Let's look at each . We can distinguish them by the value of .
Note that in we have that adds to the coefficient of if and only if .
This means that actually
But and we know how to sum this!