# Mobius function

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

\[\sum_{d|n}\mu(d)F(\frac{n}{d})=\sum_{d|n}\mu(d)\sum_{k|n/d}f(k)=\sum_{k|n}\sum_{d|n/k}\mu(d)f(k) =\sum_{k|n}f(k)\sum_{d|n/d}\mu(d)= f(n)\] (Error compiling LaTeX. ! Missing $ inserted.)

.

The Mobius function is also closely related to the Riemann zeta function, as