Difference between revisions of "Riemann xi"

From specialfunctionswiki
Jump to: navigation, search
Line 3: Line 3:
 
where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Riemann zeta function]].
 
where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Riemann zeta function]].
  
[[File:Complex Riemann Xi.jpg|500px]]
+
<div align="center">
 +
<gallery>
 +
File:Complex Riemann Xi.jpg|Domain coloring of $\xi$.
 +
</gallery>
 +
</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
+
[[Functional equation for Riemann xi]]<br />
<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>
 
  
<div class="toccolours mw-collapsible mw-collapsed">
+
=References=
<strong>Theorem:</strong> The following series expansion holds:
+
* {{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 $(7)$
$$\dfrac{d}{dz} \log \xi\left( \dfrac{-z}{1-z} \right) = \displaystyle\sum_{k=0}^{\infty} \lambda_{k+1}z^k,$$
 
where
 
$$\lambda_k = \dfrac{1}{(n-1)!} \dfrac{d^k}{ds^k} \left[ s^{k-1} \log \xi(s) \right] \Bigg|_{s=1} = \displaystyle\sum_{\rho} \left[ 1 - \left(1 - \dfrac{1}{\rho} \right)^n \right],$$
 
where this sum is over $\rho$, the non-trivial zeros of the [[Riemann zeta function]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Revision as of 15:19, 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 $\Gamma$ denotes the gamma function and $\zeta$ denotes the Riemann zeta function.

Properties

Functional equation for Riemann xi

References