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 functions Ai and Bi

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 sequence of values of x for which Ai(x) = 0, sorted by their absolute values, are called the Airy zeros and are denoted a₁, a₂, ...

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 Γ 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

$\zeta_{\mathrm{Ai}}(1)=\frac{-3^{-2/3}\Gamma(\frac23)}{\Gamma(\frac43)}.$

References

• Crandall, Richard E. (1996), "On the quantum zeta function", Journal of Physics. A. Mathematical and General 29 (21): 6795–6816, doi:10.1088/0305-4470/29/21/014, ISSN 0305-4470, MR 1421901

External links

• Weisstein, Eric W., "Airy Zeta Function", MathWorld.