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[edit]
The Airy function
is positive for positive x, but oscillates for negative values of x. The Airy zeros are the values at which , ordered by increasing magnitude: .
The Airy zeta function is the function defined from this sequence of zeros by the series
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[edit]
Like the Riemann zeta function, whose value is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2:
where 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}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/102ccd15ae853cfe89ce46a42fe11ae6fad0ee6e)
References[edit]
- 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