Define a function $\displaystyle \mu$ on the set of non-negative integers as follows. Let $\displaystyle \mu(1)=1$, and let $\displaystyle \mu(n)=0$ if n > 1 and n is divisible by the square of an integer a > 1. Otherwise, if $\displaystyle n=p_1p_2...p_k$, where the $\displaystyle p_i$ are all distinct primes, then let $\displaystyle \mu(n)=(-1)^k$. Use induction to prove that for all positive integers n > 1,

$\displaystyle Z_n=\sum_{d>0, d|n}\mu(d)=0$