# Analytic Continuations of Riemann Zeta Function

## Theorem

### Right half-plane

The Riemann zeta function has a unique analytic continuation to $\{s \in \C : \Re(s) > 0\}\setminus\{1\}$, the half-plane $\Re(s)>0$ minus the point $s=1$.

### Complex plane

The Riemann zeta function $\zeta$ has a unique analytic continuation to $\C\setminus\{1\}$.