Difference between revisions of "Riemann xi"
From specialfunctionswiki
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[[File:Complex Riemann Xi.jpg|500px]] | [[File:Complex Riemann Xi.jpg|500px]] | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The values of $\xi$ are known at even integers: | ||
| + | $$\xi(2n) = \dfrac{(-1)^{n+1}}{(2n)!}B_{2n}2^{2n-1}\pi^n (2n^2-n)(n-1)!,$$ | ||
| + | where $B_n$ is the $n$th [[Bernoulli number]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 17:36, 19 February 2015
The Riemann $\xi$ function is defined by the formula $$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$ where $\Gamma$ denotes the gamma function and $\zeta$ denotes the Riemann zeta function.
Properties
Theorem: The values of $\xi$ are known at even integers: $$\xi(2n) = \dfrac{(-1)^{n+1}}{(2n)!}B_{2n}2^{2n-1}\pi^n (2n^2-n)(n-1)!,$$ where $B_n$ is the $n$th Bernoulli number.
Proof: █
