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:

$\ds \pi^{-s/2 } \map \Gamma {\dfrac s 2} \map \zeta s = \pi^{\paren {s/2 - 1/2 } } \map \Gamma {\dfrac {1 - s} 2} \map \zeta {1 - s}$

where $\Gamma$ is the gamma function


Proof

Let $\ds \map \omega x = \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}$.

Then from Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function we have:

$(1): \quad \ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = -\frac 1 {s \paren {1 - s} } + \int_1^\infty \paren {x^{s / 2 - 1} + x^{- s / 2 - 1 / 2} } \map \omega x \rd x$


We observe that this integral is invariant under $s \mapsto 1 - s$.

Then:

\(\ds \pi^{-\paren {1 - s } / 2} \map \Gamma {\frac {1 - s} 2} \map \zeta {1 - s}\) \(=\) \(\ds -\frac 1 {\paren {1 - s} \paren {1 - \paren {1 - s} } } + \int_1^\infty \paren {x^{\paren {1 - s} / 2 - 1} + x^{-\paren {1 - s} / 2 - 1 / 2} } \map \omega x \rd x\) setting $s \mapsto 1 - s$
\(\ds \) \(=\) \(\ds - \frac 1 {\paren {1 - s} s} + \int_1^\infty \paren {x^{- s / 2 - 1 / 2 } + x^{s / 2 - 1} } \map \omega x \rd x\)
\(\ds \) \(=\) \(\ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s\) from $(1)$

as required.

$\blacksquare$


Also see


Sources