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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