Let $s=\sigma+i t$ with $\sigma, t \in \mathbb{R}$. Define the Riemann zeta function $\zeta(s)$ for $\sigma>1$. Show that for $\sigma>1$,

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

where the product is taken over all primes. Deduce that there are infinitely many primes.