# Analytic Continuations of Riemann Zeta Function to Complex Plane

## Theorem

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

## Proof

Note that by Riemann Zeta Function is Analytic, $\zeta(s)$ is indeed analytic for $\Re(s)>1$.

By Complex Plane minus Point is Connected, $\C\setminus\{1\}$ is connected.

By Uniqueness of Analytic Continuation, there is at most one analytic continuation of $\zeta$ to $\C\setminus\{1\}$.