Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function

Theorem
Let $\zeta$ be the Riemann zeta function.

Let $\eta$ be the Dirichlet Eta Function.

Then:
 * $\dfrac 1 {1 - 2^{1 - s} } \map \eta s$

defines an analytic continuation of $\zeta$ to the half-plane $\map \Re s > 0$ minus $s = 1$.

Proof
By Riemann Zeta Function in terms of Dirichlet Eta Function, it coincides with $\zeta$ for $\map \Re s > 1$.

By Dirichlet Eta Function is Analytic, it is analytic for $\map \Re s > 0$, except at $s = 1$.

Also see

 * Analytic Continuations of Riemann Zeta Function to Right Half-Plane