Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function

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Theorem

Let $\zeta$ be the Riemann zeta function.


Then

$\displaystyle\frac{\pi^{s/2}}{\Gamma \left({\frac s 2}\right)} \cdot\left( - \frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left({x^{s / 2 - 1} + x^{- \left({s + 1}\right) / 2} }\right) \omega \left({x}\right) \ \mathrm d x \right)$

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