# Functional Equation for Riemann Zeta Function

Jump to navigation
Jump to search

## Contents

## 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 35$: Riemann Zeta Function $\displaystyle \map \zeta x = \frac 1 {1^x} + \frac 1 {2^x} + \frac 1 {3^x} + \ldots$: $35.25$