Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function

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

Then
 * $\ds \frac {\pi^{s/2}} {\map \Gamma {\frac s 2}} \cdot \paren{ - \frac 1 {s \paren{1 - s}} + \int_1^\infty \paren{x^{s / 2 - 1} + x^{- \paren{s + 1} / 2} } \map \omega x \ \mathrm d x}$

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

Proof
By Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function, it coincides with $\zeta(s)$ for $\Re(s)>1$.

Interchanging integral and derivative, one shows that the integral is analytic for $\Re(s)>0$.

Also see

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