Möbius inversion formula

The Möbius inversion formula in number theory states that if F(n) and f(n) are arithmetic functions satisfying

F(n) = ∑_d|nf(d)

then

f(n) = ∑_d|n μ(d)F(n/d)

where μ(d) represents the Möbius function and all sums extend over all positive divisors of n.

With this function for example it can be easily shown that Euler's function φ(n) is multiplicative.

In this way every arithmetic function can therefore generate another one.