Example 4 Find the Möbius inverse transform $f(n)$ of $F(n)=n^{t}$.


Solve $n^{t}$ is multiplicative, from equation (27) we get
$$f\left(p^{a}\right)=p^{a t}-p^{(\sigma-1) t}=p^{a t}\left(1-p^{-t}\right) .$$

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