(a) What does it mean to say that a function $f: \mathbb{N} \rightarrow \mathbb{C}$ is multiplicative? Show that if $f, g: \mathbb{N} \rightarrow \mathbb{C}$ are both multiplicative, then so is $f \star g: \mathbb{N} \rightarrow \mathbb{C}$, defined for all $n \in \mathbb{N}$ by

$$f \star g(n)=\sum_{d \mid n} f(d) g\left(\frac{n}{d}\right)$$

Show that if $f=\mu \star g$, where $\mu$ is the Möbius function, then $g=f \star 1$, where 1 denotes the constant function 1 .

(b) Let $\tau(n)$ denote the number of positive divisors of $n$. Find $f, g: \mathbb{N} \rightarrow \mathbb{C}$ such that $\tau=f \star g$, and deduce that $\tau$ is multiplicative. Hence or otherwise show that for all $s \in \mathbb{C}$ with $\operatorname{Re}(s)>1$,

$$\sum_{n=1}^{\infty} \frac{\tau(n)}{n^{s}}=\zeta(s)^{2}$$

where $\zeta$ is the Riemann zeta function.

(c) Fix $k \in \mathbb{N}$. By considering suitable powers of the product of the first $k+1$ primes, show that

$$\tau(n) \geqslant(\log n)^{k}$$

for infinitely many $n \in \mathbb{N}$.

(d) Fix $\epsilon>0$. Show that

$$\frac{\tau(n)}{n^{\epsilon}}=\prod_{p \text { prime, } p^{\alpha}|| n} \frac{(\alpha+1)}{p^{\alpha \epsilon}}$$

where $p^{\alpha} \| n$ denotes the fact that $\alpha \in \mathbb{N}$ is such that $p^{\alpha} \mid n$ but $p^{\alpha+1} \nmid n$. Deduce that there exists a positive constant $C(\epsilon)$ depending only on $\epsilon$ such that for all $n \in \mathbb{N}, \tau(n) \leqslant C(\epsilon) n^{\epsilon} .$