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Airy Zeta Function

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AiryZeta

Define the Airy zeta function for n=2, 3, ... by

Z(n)=sum_(r)1/(r^n),
(1)

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

Z(n)=(piT_(n-1)(0))/(Gamma(n)),
(2)

where Gamma(z) is the gamma function,

T_n(z)=C^((n))(z)A+(d^(n-1))/(dz^(n-1))[Ai(z)Bi(z)] -sum_(j=1)^n(n; j)C^((n-j))(z)(d^(j-1))/(dz^(j-1))[Ai(z)]^2,
(3)

where

A=int_0^infty[Ai(z)]^2dz
(4)
=1/(3^(2/3)[Gamma(1/3)]^2)
(5)

and

C(z)=(Bi(z))/(Ai(z))
(6)

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

Surprisingly, defining

X=1/(2piAi(0)Bi(0))
(7)
=(3^(5/6))/(2pi)[Gamma(2/3)]^2
(8)
=(2pi)/(3^(1/6)[Gamma(1/3)]^2)
(9)

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

Z(2)=X^2
(10)
Z(3)=1/2(2X^3-1)
(11)
Z(4)=1/3(3X^4-X)
(12)
Z(5)=1/(12)(12X^5-5X^2)
(13)
Z(6)=1/(20)(20X^6-10X^3+1)
(14)

(OEIS A096631 and A096632). The corresponding numerical values are approximately 0.531457, -0.112562, 0.0394431, -0.0155337, and 0.00638927, ....

See also

Airy Functions, Zeta Function

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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."

Referenced on Wolfram|Alpha

Airy Zeta Function

Cite this as:

Weisstein, Eric W. "Airy Zeta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AiryZetaFunction.html

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