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: | 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: | ||
$$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$ | $$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$ | ||
| − | + | ||
| − | $$\zeta_{\mathrm{Ai}}(2) = \dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$ | + | =Properties= |
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> The following formula holds: | ||
| + | $$\zeta_{\mathrm{Ai}}(2)=\dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
=References= | =References= | ||
[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)] | ||
Revision as of 23:16, 1 April 2015
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)
