Hankel Representation of Riemann Zeta Function

Theorem

Let $C$ be the Hankel contour.

Then for $s \in \C \setminus \Z_{>0}$:

$\ds \map \zeta s = \frac {i \Gamma \paren {1 - s} } {2 \pi} \oint_C \frac {\paren {-z}^{s - 1} } {e^z - 1} \rd z$

where:

$\zeta$ is the Riemann Zeta function

$\Gamma$ is the Gamma function.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

This entry was named for Hermann Hankel.

Sources

• November 1859: Bernhard Riemann: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse ("On the Number of Primes Less Than a Given Magnitude") (Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin)