Functional Equation for Riemann Zeta Function

Theorem

Let $\zeta$ be the Riemann zeta function.

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

Then:

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

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

Functional Equation for Completed Riemann Zeta Function

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

Sources

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1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Riemann Zeta Function $\map \zeta x = \dfrac 1 {1^x} + \dfrac 1 {2^x} + \dfrac 1 {3^x} + \cdots$: $35.25$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 36$: Miscellaneous and Riemann Zeta Functions: Riemann Zeta Function $\map \zeta x = \dfrac 1 {1^x} + \dfrac 1 {2^x} + \dfrac 1 {3^x} + \cdots$: $36.25.$