For $s \in \mathbb{C}$ and $\sigma = \Re(s)>1$, $$\zeta(s) = \prod\limits_{p \in \mathbb{P}}\left(1 - \frac{1}{p^s}\right)^{-1}$$
My question is: is the above correct? Or should the $s$ be replaced with $\sigma$?
Also: For all $s \in \mathbb{C}$ with $\Re(s)>1$, $\zeta(s) \neq 0$.
Proof:
For $\sigma=\Re(s)>1$, $$\frac{1}{|\zeta(s)|} = \left|\prod\limits_{p \in \mathbb{P}}\left(1 - \frac{1}{p^s}\right)\right| \leq \prod\limits_{p \in \mathbb{P}}\left(1 + \frac{1}{p^{\sigma}}\right) < \sum_{n=1}^{\infty} \frac{1}{n^{\sigma}} \leq1+ \int_1^{\infty} \frac{du}{u^{\sigma}} = \frac{\sigma}{\sigma -1} $$ Hence $|\zeta(s)| > \frac{\sigma-1}{\sigma}>0$.
I don't get why the inequalities hold. I can see the integral is from the integral test.