Example 6 Find the Möbius inverse transform $f(n)$ of $F(n)=\Lambda(n)$.


$\Lambda(n)$ is not multiplicative, so we cannot use equation (27). However, using equation (19) to obtain a good expression for $f(n)$ is quite difficult. Noting equation (24), we have
$$\Lambda(n)=\left(\sum_{d \mid n} \mu(d)\right) \ln n-\sum_{d \mid n} \mu(d) \ln d$$

Using equation (18), we get
$$\Lambda(n)=-\sum_{d \mid n} \mu(d) \ln d=\sum_{d \mid n} \mu(d) \ln \frac{1}{d}$$

This proves that
$$f(n)=\mu(n) \ln (1 / n)=-\mu(n) \ln n$$