A “Möbius” inversion formula
[Not the Möbius inversion formula, but something similar.]
As usual, define the Möbius function μ on the natural numbers as
Let be any function defined on the natural numbers, and let be the function defined as .
Then it is true that .
Note that f need not be multiplicative; it can be any arbitrarily defined function. I have no idea why it is true. Help?