Airy Zeta Function

Define the Airy zeta function for , 3, ... by

(1)

where the sum is over the real (negative) zeros  of the Airy function . This has the closed-form representation

(2)

where  is the gamma function,

(3)

where

			(4)
			(5)

and

(6)

(Crandall 1996; Borwein et al. 2004, p. 61).

Surprisingly, defining

			(7)
			(8)
			(9)

gives  as a polynomial in  (Borwein et al. 2004, pp. 61-62). The first few such polynomials are

			(10)
			(11)
			(12)
			(13)
			(14)

(OEIS A096631 and A096632). The corresponding numerical values are approximately 0.531457, , 0.0394431, , and 0.00638927, ....

REFERENCES:

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 61-62, 2004.

Crandall, R. E. "On the Quantum Zeta Function." J. Phys. A: Math. General 29, 6795-6816, 1996.

Sloane, N. J. A. Sequences A096631 and A096632 in "The On-Line Encyclopedia of Integer Sequences."