Example 5 Find the Möbius inverse transform of $F(n)=\varphi(n)$.
Since $\varphi(n)$ is multiplicative, from equation (27) we get
$$f\left(p^{\alpha}\right)=\varphi\left(p^{\alpha}\right)-\varphi\left(p^{\sigma-1}\right)=\left\{\begin{array}{ll}
p(1-2 / p), & \alpha=1 \\
p^{\alpha}\left(1-p^{-1}\right)^{2}, & \alpha \geqslant 2 .
\end{array}\right.$$

Therefore, we have
$$f(n)=n \prod_{p \| n}\left(1-\frac{2}{p}\right) \prod_{p^{2} \mid n}\left(1-\frac{1}{p}\right)^{2}$$