Difference between revisions of "Airy zeta function"
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| − | The [[Airy | + | 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: |
$$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$ | $$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$ | ||
Revision as of 17:40, 25 March 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)
