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

File:Airy Functions.svg

The Airy functions Ai and Bi

The Airy function

$\mathrm{A}\mathrm{i}(x)=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\cos (\frac{1}{3}{t}^3+xt)dt,$

is positive for positive x, but oscillates for negative values of x. The Airy zeros are the values $\{a_i{\}}_{i=1}^{\mathrm{\infty }}$ 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

${\zeta }_{\mathrm{A}\mathrm{i}}(s)=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{|a_i{|}^s}.$

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:

${\zeta }_{\mathrm{A}\mathrm{i}}(2)=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{a_i^2}=\frac{3^{5/3}{\Gamma }^4(\frac{2}{3})}{4{\pi }^2},$

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

${\zeta }_{\mathrm{A}\mathrm{i}}(1)=-\frac{\Gamma (\frac{2}{3})}{\Gamma (\frac{4}{3})\sqrt[3]{9}}.$