Analytic Continuations of Riemann Zeta Function to Right Half-Plane
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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:
- 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
there exists one.
$\blacksquare$