Functional Equation for Riemann Zeta Function

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Theorem

Let $\zeta$ be the Riemann zeta function.

Let $\map \zeta s$ have an analytic continuation for $\map \Re s > 0$.


Then:

$\displaystyle \map \Gamma {\frac s 2} \pi^{-s/2} \map \zeta s = \map \Gamma {\frac {1 - s} 2} \pi^{\frac {s - 1} 2} \map \zeta {1 - s}$

where $\Gamma$ is the gamma function


Proof

Using Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function:

The functional equation:

$\map \xi s = \map \xi {1 - s}$

follows upon observing that this integral is invariant under $s \mapsto 1 - s$.



Also see


Sources