Difference between revisions of "Airy zeta function"

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The [[Airy functions | Airy function]] $\mathrm{Ai}$ is oscillatory for negative values of $x$. This yields a sequence of zeros $\{a_i\}_{i=1}^{\infty}$. We define the Airy zeta function using these zeros in the following way:
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The [[Airy Ai | Airy function]] $\mathrm{Ai}$ is oscillatory for negative values of $x$. This yields a sequence of zeros $\{a_i\}_{i=1}^{\infty}$. We define the Airy zeta function using these zeros in the following way:
 
$$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$
 
$$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$
  
 
=Properties=
 
=Properties=
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[[Airy zeta function at 2]]<br />
<strong>Proposition:</strong> The following formula holds:
 
$$\zeta_{\mathrm{Ai}}(2)=\dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$
 
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<strong>Proof:</strong>  █
 
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=References=
 
=References=
[http://en.wikipedia.org/wiki/Airy_zeta_function Airy zeta function (Wikipedia)]<br />
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* {{PaperReference|On the quantum zeta function|1996|Richard E. Crandall}}
[http://mathworld.wolfram.com/AiryZetaFunction.html Airy zeta function (Mathworld)]
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[[Category:SpecialFunction]]

Latest revision as of 02:16, 2 November 2016

The Airy function $\mathrm{Ai}$ is oscillatory for negative values of $x$. This yields a sequence of zeros $\{a_i\}_{i=1}^{\infty}$. We define the Airy zeta function using these zeros in the following way: $$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$

Properties

Airy zeta function at 2

References