Airy zeta function
The Airy function
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
Evaluation at integers
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:
It is conjectured that the analytic continuation of the Airy zeta function evaluates at 1 to
- 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