Analytic Continuations of Riemann Zeta Function to Right Half-Plane

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

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

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

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

By either: there exists one.
 * Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function
 * Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part
 * Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function

Also see

 * Analytic Continuations of Riemann Zeta Function