Example 2 Find the Möbius transform of $\mu^{2}(n) / \varphi(n)$.

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Solve $\sum_{d \mid p^{e^{\circ}}} \mu^{2}(d) / \varphi(d)=1+1 /(p-1)=(1-1 / p)^{-1}$.
From this and the fact that $\mu^{2}(n) / \varphi(n)$ is a multiplicative function, we get
$$\sum_{d \mid n} \mu^{2}(d) / \varphi(d)=\prod_{p \mid n}(1-1 / p)^{-1}=n / \varphi(n)$$