## Monday Math 86

A notable multiplicative function is the Möbius function μ(n) defined as follows:
μ(n)=1 when n is a square-free integer with an even number of distinct prime factors
μ(n)=-1 when n is a square-free integer with an odd number of distinct prime factors
μ(n)=0 when n is not square-free.
Note that μ(1)=1, as 1 has an even number of prime factors; namely, zero.
In terms of the distinct prime factor counting function ω(n) (see here), we can write the Möbius function as:

One can easily confirm that μ(n) is a multiplicative function (but not a completely multiplicative function); if either m or n is not square-free (or both), then mn is not square-free [and μ(mn)=μ(m)μ(n)=0]. If m and n are both square-free and coprime, they have no common prime factors, so mn is square-free and ω(mn)=ω(m)+ω(n) [so ].
In terms of the powers of primes, we see:
.