Analytic Continuations of Riemann Zeta Function to Complex Plane

Theorem

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

Proof

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

By Complex Plane minus Point is Connected, $\C\setminus\{1\}$ is connected.

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

This needs considerable tedious hard slog to complete it. In particular: link to proofs of various analytic continuations To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

• Functional Equation for Riemann Zeta Function, analyticity is shown at Analytic Continuation of Riemann Zeta Function
• The stepwise method as in Riemann Zeta Function in terms of Dirichlet Eta Function

Also see

• Analytic Continuations of Riemann Zeta Function