Difference between revisions of "Airy zeta function"

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[http://en.wikipedia.org/wiki/Airy_zeta_function Airy zeta function (Wikipedia)]<br />
 
[http://en.wikipedia.org/wiki/Airy_zeta_function Airy zeta function (Wikipedia)]<br />
 
[http://mathworld.wolfram.com/AiryZetaFunction.html Airy zeta function (Mathworld)]
 
[http://mathworld.wolfram.com/AiryZetaFunction.html Airy zeta function (Mathworld)]
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[[Category:SpecialFunction]]

Revision as of 18:51, 24 May 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

Proposition: The following formula holds: $$\zeta_{\mathrm{Ai}}(2)=\dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$

Proof:

References

Airy zeta function (Wikipedia)
Airy zeta function (Mathworld)