(i) Let $\pi(x)$ denote the number of primes $\leqslant x$, where $x$ is a positive real number. State and prove Legendre's formula relating $\pi(x)$ to $\pi(\sqrt{x})$. Use this formula to compute $\pi(100)-\pi(10) .$

(ii) Let $\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$, where $s$ is a real number greater than 1 . Prove the following two assertions rigorously, assuming always that $s>1$. (a) $\zeta(s)=\prod_{p}\left(1-p^{-s}\right)^{-1}$, where the product is taken over all primes $p$; (b) $\zeta(s)=\frac{1}{1-2^{1-s}} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}$.

Explain why (b) enables us to define $\zeta(s)$ for $0<s<1$. Deduce from (b) that $\lim _{s \rightarrow 1}(s-1) \zeta(s)=1$.