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.

Definition

The Airy functions Ai and Bi

The Airy function

{\mathrm {Ai}}(x)={\frac {1}{\pi }}\int _{0}^{\infty }\cos \left({\tfrac 13}t^{3}+xt\right)\,dt,

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

\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 $\zeta (2)=\pi ^{2}/6$ is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2:

\zeta _{{{\mathrm {Ai}}}}(2)=\sum _{{i=1}}^{{\infty }}{\frac {1}{a_{i}^{2}}}={\frac {3^{{5/3}}\Gamma ^{4}({\frac 23})}{4\pi ^{2}}},

where Γ 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

\zeta _{{{\mathrm {Ai}}}}(1)={\frac {-3^{{-2/3}}\Gamma ({\frac 23})}{\Gamma ({\frac 43})}}.

References

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

External links

Weisstein, Eric W. "Airy Zeta Function". MathWorld.

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