Let $f: \mathbb{N} \rightarrow \mathbb{R}$ be a function, where $\mathbb{N}$ denotes the (positive) natural numbers.

Define what it means for $f$ to be a multiplicative function.

Prove that if $f$ is a multiplicative function, then the function $g: \mathbb{N} \rightarrow \mathbb{R}$ defined by

$$g(n)=\sum_{d \mid n} f(d)$$

is also multiplicative.

Define the Möbius function $\mu$. Is $\mu$ multiplicative? Briefly justify your answer.

Compute

$$\sum_{d \mid n} \mu(d)$$

for all positive integers $n$.

Define the Riemann zeta function $\zeta$ for complex numbers $s$ with $\Re(s)>1$.

Prove that if $s$ is a complex number with $\Re(s)>1$, then

$$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}$$