8. Find the Möbius transform and the Möbius inverse transform of $|\mu(n)|$.


8. $\sum_{d \mid n}|\mu(n)|=2^{\alpha(n)}, \sum_{d \mid n} \mu(d)|\mu(n / d)|=1$, when $n=1 ;=0$, when $n=p$ or $p^{\alpha}$, $\alpha>2 ;=-1$, when $n=p^{2}$, i.e.
$$\sum_{d \mid n} \mu(d)|\mu(n / d)|=\left\{\begin{array}{ll}
1, & n=1 \\
(-1)^{r}, & n=p_{1}^{2} \cdots p_{r}^{2}, \\
0, & \text { otherwise. }
\end{array}\right.$$