Let $\zeta(s)$ be the Riemann zeta function, and put $s=\sigma+i t$ with $\sigma, t \in \mathbb{R}$.

(i) If $\sigma>1$, prove that

$$\zeta(s)=\prod_{p}\left(1-p^{-s}\right)^{-1}$$

where the product is taken over all primes $p$.

(ii) Assuming that, for $\sigma>1$, we have

$$\zeta(s)=\sum_{n=1}^{\infty} n\left(n^{-s}-(n+1)^{-s}\right)$$

prove that $\zeta(s)-\frac{1}{s-1}$ has an analytic continuation to the half plane $\sigma>0$.