The Riemann zeta function $\zeta(s)$ satisfies: 1. $\zeta(s)$ is holomorphic for $\operatorname{Re}(s) > 1$. 2. The function $\zeta(s) - \frac{1}{s-1}$ extends to a holomorphic function for $\operatorname{Re}(s) > 0$. In other words, $\zeta(s)$ extends meromorphically to $\operatorname{Re}(s) > 0$ with a simple pole at $s = 1$ with residue $1$. 3. For $\operatorname{Re}(s) > 1$, the **Euler product** holds: \begin{align*} \zeta(s) = \prod_{p\text{ prime}} \left(1 - p^{-s}\right)^{-1}, \end{align*} and the product is absolutely convergent.