Airy zeta function

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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.

The Airy functions Ai and Bi

The Airy function

\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 $\{a_{i}\}_{i=1}^{\infty }$ at which ${\text{Ai}}(a_{i})=0$ , ordered by increasing magnitude: $|a_{1}|<|a_{2}|<\cdots$ .

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 {2}{3}})}{4\pi ^{2}}},

where $\Gamma$ 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

\zeta _{\mathrm {Ai} }(1)=-{\frac {\Gamma ({\frac {2}{3}})}{\Gamma ({\frac {4}{3}}){\sqrt[{3}]{9}}}}.

References

External links

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