What is a multiplicative function? Show that if $f(n)$ is a multiplicative function, then so is $g(n)=\sum_{d \mid n} f(d)$.

Define the Möbius function $\mu(n)$, and show that it is multiplicative. Deduce that

$$\sum_{d \mid n} \mu(d)= \begin{cases}1 & \text { if } n=1 \\ 0 & \text { if } n>1\end{cases}$$

and that

$$f(n)=\sum_{e \mid n} \mu(e) g\left(\frac{n}{e}\right) .$$

What is $g(n)$ if $f(n)=n ?$ What is $f(n)$ if $g(n)=n ?$