Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function

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Theorem

Let $\zeta$ be the Riemann zeta function.

Then

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

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

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Proof

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

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

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Also see

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