Difference between revisions of "Riemann xi"

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The Riemann $\xi$ function is defined by the formula
 
The Riemann $\xi$ function is defined by the formula
 
$$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$
 
$$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$
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where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Riemann zeta function]].
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where $\pi$ denotes [[pi]], $\Gamma$ denotes [[gamma]], and $\zeta$ denotes [[Riemann zeta]].
  
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[[File:Complex Riemann Xi.jpg|500px]]
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<div align="center">
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<gallery>
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File:Complex Riemann Xi.jpg|Domain coloring of $\xi$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Functional equation for Riemann xi]]<br />
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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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=References=
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where $B_n$ is the $n$th [[Bernoulli number]].
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* {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Functional equation for Riemann zeta with cosine|next=Functional equation for Riemann xi}}: § Introduction $(7)$
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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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[[Category:SpecialFunction]]
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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>
 

Latest revision as of 15:31, 18 March 2017

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 $\pi$ denotes pi, $\Gamma$ denotes gamma, and $\zeta$ denotes Riemann zeta.

  • Domain coloring of $\xi$.

Properties

Functional equation for Riemann xi

References

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