Difference between revisions of "Functional equation for Riemann zeta with cosine"
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(Created page with "==Theorem== The following formula holds for all $z \in \mathbb{C}$: $$\zeta(1-z)=2^{1-z} \pi^{-z} \cos \left( \dfrac{\pi z}{2} \right)\Gamma(z)\zeta(z),$$ where $\zeta$ denote...") |
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Revision as of 00:01, 18 March 2017
Theorem
The following formula holds for all $z \in \mathbb{C}$: $$\zeta(1-z)=2^{1-z} \pi^{-z} \cos \left( \dfrac{\pi z}{2} \right)\Gamma(z)\zeta(z),$$ where $\zeta$ denotes Riemann zeta, $\pi$ denotes pi, $\cos$ denotes cosine, and $\Gamma$ denotes gamma.
Proof
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(6')$
