Mobius Function

Definition

The mobius function $\mu$ is defined as follows:

\mu(1)=1

If $n > 1$ , write $n=p_1^{a_1} \cdots p_k^{a_k}$ . Then

\mu(n)=(-1)^k \text{ if } a_1=\cdots=a_k=1

\mu(n) = 0. \text{ otherwise}

Notice that $\mu(n) = 0$ if and only if n has a square factor > 1.

Here is a short table of values of $\mu(n)$ :

$n:$	1	2	3	4	5	6	7	8	9	10
$\mu(n):$	1	-1	-1	0	-1	1	-1	0	0	1

If $n \ge 1$ we have

\sum_{d|n}\mu(d)= \left\lfloor \frac{1}{n} \right\rfloor = \left\{ \begin{array}{cl} 1 & : \ n = 1. \\ 0 & : \ n > 1. \end{array} \right.

Prove that

\sum_{d^2|n} \mu(d) = \mu(n)^2

And more generally

\sum_{d^k|n} \mu(d) = \left\{ \begin{array}{cl} 0 & : \text{if }m^k|n \text{ for some } m>1 \\ 1& : \text{otherwise} \end{array} \right.