Analytic Continuations of Riemann Zeta Function

Theorem

Right half-plane

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 $s = 1$.

Complex plane

The Riemann zeta function $\zeta$ has a unique analytic continuation to $\C\setminus\{1\}$.

Also see

• Poles of Riemann Zeta Function