Möbius function

where the Möbius function m is defined as follows:
let a natural number n to be written as a multiplication of primes:

Then m(n) = (-1)^r where k₁= .....=k_r=1; m(1) = 1; m(n) = 0 otherwise.

It follows that m = -1 for each prime. This extraordinary function is used in the Prime Number Theorem.