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.

The Airy function

${\displaystyle \mathrm {Ai} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\cos \left({\tfrac {1}{3}}t^{3}+xt\right)\,dt,}$

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: ${\displaystyle |a_{1}|<|a_{2}|<\cdots }$ .

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.

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:

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

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

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

• Crandall, Richard E. (1996), "On the quantum zeta function", Journal of Physics A: Mathematical and General, 29 (21): 6795–6816, Bibcode:1996JPhA...29.6795C, doi:10.1088/0305-4470/29/21/014, ISSN 0305-4470, MR 1421901

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