Airy zeta function

In mathematics, the Airy zeta function, studied by, is a function analogous to the Riemann zeta function and related to the zeros of the Airy function.

Definition
The Airy function
 * $$\mathrm{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,$$

is positive for positive x, but oscillates for negative values of x. The Airy zeros are the values $$\{a_i\}_{i=1}^\infty$$ at which $$\text{Ai}(a_i) = 0$$, ordered by increasing magnitude: $$|a_1|<|a_2|<\cdots$$.

The Airy zeta function is the function defined from this sequence of zeros by the series
 * $$\zeta_{\mathrm{Ai}}(s)=\sum_{i=1}^{\infty} \frac{1}{|a_i|^s}.$$

This series converges when the real part of s is greater than 3/2, and may be extended by analytic continuation to other values of s.

Evaluation at integers
Like the Riemann zeta function, whose value $$\zeta(2)=\pi^2/6$$ is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2:
 * $$\zeta_{\mathrm{Ai}}(2)=\sum_{i=1}^{\infty} \frac{1}{a_i^2}=\frac{3^{5/3}\Gamma^4(\frac23)}{4\pi^2},$$

where $$\Gamma$$ is the gamma function, a continuous variant of the factorial. Similar evaluations are also possible for larger integer values of s.

It is conjectured that the analytic continuation of the Airy zeta function evaluates at 1 to
 * $$\zeta_{\mathrm{Ai}}(1)=-\frac{\Gamma(\frac23)}{\Gamma(\frac43)\sqrt[3]{9}}.$$