Example 3 Find the Möbius transform $F(n)$ of $\Omega(n)$.


Solve $\Omega(n)$ is not a multiplicative function. Only formula (2) can be used to calculate. $F(1)=0,1<n=$ $p_{1}^{a_{1}} \cdots p_{r}^{\sigma}$ when,
$$\begin{array}{l}
\quad F(n)=\sum_{e_{1}=0}^{\alpha_{1}} \cdots \sum_{e_{r}=0}^{\alpha_{r}} \Omega\left(p_{1}^{f_{1}} \cdots p_{r}^{e_{r}}\right) \\
=\sum_{e_{1}=0}^{\alpha_{1}} \cdots \sum_{e_{r}=0}^{\alpha_{r}}\left(e_{1}+\cdots+e_{r}\right) \\
=\frac{1}{2} \alpha_{1}\left(\alpha_{1}+1\right) \cdots\left(\alpha_{r}+1\right)+\cdots+\frac{1}{2} \alpha_{r}\left(\alpha_{1}+1\right) \cdots\left(\alpha_{r}+1\right) \\
=\frac{1}{2} \Omega(n) \tau(n) .
\end{array}$$