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Airy zeta function

From Wikipedia, the free encyclopedia

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

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The Airy functions Ai and Bi

The Airy function

is positive for positive x, but oscillates for negative values of x. The Airy zeros are the values at which , ordered by increasing magnitude: .

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

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

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Like the Riemann zeta function, whose value is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2:

where 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

References

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  • Crandall, Richard E. (1996), "On the quantum zeta function", Journal of Physics A: Mathematical and General, 29 (21): 6795–6816, Bibcode:1996JPhA...29.6795C, doi:10.1088/0305-4470/29/21/014, ISSN 0305-4470, MR 1421901
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