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| | [[File:Complex Riemann Xi.jpg|500px]] | | [[File:Complex Riemann Xi.jpg|500px]] |
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| − | =Properties=
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| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> The values of $\xi$ are known at even integers:
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| − | $$\xi(2n) = \dfrac{(-1)^{n+1}}{(2n)!}B_{2n}2^{2n-1}\pi^n (2n^2-n)(n-1)!,$$
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| − | where $B_n$ is the $n$th [[Bernoulli number]].
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> The following series expansion holds:
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| − | $$\dfrac{d}{dz} \log \xi\left( \dfrac{-z}{1-z} \right) = \displaystyle\sum_{k=0}^{\infty} \lambda_{k+1}z^k,$$
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| − | where
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| − | $$\lambda_k = \dfrac{1}{(n-1)!} \dfrac{d^k}{ds^k} \left[ s^{k-1} \log \xi(s) \right] \Bigg|_{s=1} = \displaystyle\sum_{\rho} \left[ 1 - \left(1 - \dfrac{1}{\rho} \right)^n \right],$$
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| − | where this sum is over $\rho$, the non-trivial zeros of the [[Riemann zeta function]].
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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Revision as of 20:33, 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.
