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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.

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Definition

The Airy function

A i ( x ) = 1 π ∫ 0 ∞ cos ⁡ ( 1 3 t 3 + x t ) d t , {\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 { a i } i = 1 ∞ {\displaystyle \{a_{i}\}_{i=1}^{\infty }} at which Ai ( a i ) = 0 {\displaystyle {\text{Ai}}(a_{i})=0} , ordered by increasing magnitude: | a 1 | < | a 2 | < ⋯ {\displaystyle |a_{1}|<|a_{2}|<\cdots } .

The Airy zeta function is the function defined from this sequence of zeros by the series

ζ A i ( s ) = ∑ i = 1 ∞ 1 | a i | s . {\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.

Evaluation at integers

Like the Riemann zeta function, whose value ζ ( 2 ) = π 2 / 6 {\displaystyle \zeta (2)=\pi ^{2}/6} is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2:

ζ A i ( 2 ) = ∑ i = 1 ∞ 1 a i 2 = 3 5 / 3 Γ 4 ( 2 3 ) 4 π 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 Γ {\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

ζ A i ( 1 ) = − Γ ( 2 3 ) Γ ( 4 3 ) 9 3 . {\displaystyle \zeta _{\mathrm {Ai} }(1)=-{\frac {\Gamma ({\frac {2}{3}})}{\Gamma ({\frac {4}{3}}){\sqrt[{3}]{9}}}}.}