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 $\C\setminus\{1\}$.


 * Functional Equation for Riemann Zeta Function, analyticity is shown at Analytic Continuation of Riemann Zeta Function
 * The stepwise method as in Riemann Zeta Function in terms of Dirichlet Eta Function

Also see

 * Analytic Continuations of Riemann Zeta Function