The Mobius function is a multiplicative number theoretic function defined as follows: In addition, .
The Mobius function is useful for a variety of reasons.
First, it conveniently encodes Principle of Inclusion-Exclusion. For example, to count the number of positive integers less than or equal to and relatively prime to , we have
more succinctly expressed as
One unique fact about the Mobius function, which leads to the Mobius inversion formula, is that
Property 1: The function is multiplicative .
Proof:If or for a prime , we are done.Else let and where ,then .
Property 2:If for every positive integer , then . Proof:We have .
The Mobius function is also closely related to the Riemann zeta function, as