Example 1 Find the Möbius transform of the Liouville function $\lambda(n)$.
$$\text{Solve } \begin{aligned}
\sum_{d \mid \varepsilon_{e}} \lambda(d) & =(-1)^{0}+(-1)^{1}+\cdots+(-1)^{\alpha} \\
& =\left\{\begin{array}{ll}
1, & 2 \mid \alpha, \\
0, & 2 \nmid \alpha .
\end{array}\right.
\end{aligned}$$

From this and the fact that $\lambda(n)$ is a multiplicative function, we get
$$\sum_{d \mid n} \lambda(d)=\left\{\begin{array}{ll}
1, & n \text{ is a perfect square, } \\
0, & \text{ otherwise. }
\end{array}\right.$$