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 $\{s \in \C : \Re(s) > 0\}\setminus\{1\}$, 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 Half-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:

there exists one.

$\blacksquare$


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