Analytic Continuations of Riemann Zeta Function to Right Half-Plane

Theorem
The Riemann zeta function has a unique analytic continuation to the half-plane $\Re(s)>0$ minus the point $s=1$.

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

By Complex Helf-Plane minus Point is Connected, $\{\sigma>0\}\setminus\{1\}$ is connected.

By Uniqueness of Analytic Continuation, there is at most one analytic continuation of $\zeta$ to $\{\sigma>0\}\setminus\{1\}$.

By either:
 * Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part

there exists one.


 * Riemann Zeta Function in terms of Dirichlet Eta Function
 * Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function

Also see

 * Poles of Riemann Zeta Function
 * Analytic Continuations of Riemann Zeta Function