# Airy zeta function

In mathematics, the Airy zeta function, studied by (Crandall 1996), is a function analogous to the Riemann zeta function and related to the zeros of the Airy function.

## Definition

The Airy function

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

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

The Airy zeta function is the function defined from this sequence of zeros by the series

$\displaystyle{ \zeta_{\mathrm{Ai}}(s)=\sum_{i=1}^{\infty} \frac{1}{|a_i|^s}. }$

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 $\displaystyle{ \zeta(2)=\pi^2/6 }$ is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2:

$\displaystyle{ \zeta_{\mathrm{Ai}}(2)=\sum_{i=1}^{\infty} \frac{1}{a_i^2}=\frac{3^{5/3}\Gamma^4(\frac23)}{4\pi^2}, }$

where $\displaystyle{ \Gamma }$ 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

$\displaystyle{ \zeta_{\mathrm{Ai}}(1)=-\frac{\Gamma(\frac23)}{\Gamma(\frac43)\sqrt{9}}. }$