Euler product for the Riemann zeta function (reviewed)

The Euler product for the Riemann zeta function is its expansion as the infinite product $$\zeta(s) = \prod_{p \textrm{ prime}} \frac{1}{1-p^{-s}}. $$ This expansion holds in the region $\Re(s)>1,$ where the product is absolutely convergent. The first proof of this expansion, due to Euler (1737), is based on the fact that every positive integer $n$ can be written uniquely as a product of primes. Namely, for $\Re(s) >1,$

$$ \prod_{p } \frac{1}{1-p^{-s}} = \prod_p \left(\sum_{k=0}^\infty p^{-ks} \right) = \sum_{n=1}^\infty n^{-s} = \zeta(s).$$