Mobius function
The Mobius function is a multiplicative number theoretic function defined as follows:
![\[\mu(n) = \begin{cases} 0 & d^2 | n, \\ (-1)^k & n = p_1p_2\cdots{p_k} .\end{cases}\]](http://latex.artofproblemsolving.com/7/f/5/7f54ff985c575ba980c7a2a4e45a26dd2af7481f.png)
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
\begin{align*}
\phi(n) = &n
&- \frac{n}{p_1} - \frac{n}{p_2} - \cdots - \frac{n}{p_k}
&+ \frac{n}{p_1p_2} + \frac{n}{p_1p_3} + \cdots + \frac{n}{p_{k-1}p_k}
&\vdots
&+ (-1)^k \frac{n}{p_1p_2\cdots p_k},
\end{align*}
more succinctly expressed as
![\[\phi(n) = \sum_{d|n} d \mu\left(\frac{n}{d}\right).\]](http://latex.artofproblemsolving.com/7/4/5/745b1c3b13454ca1dfc6b7a9728ac1c689fc2095.png)
