Airy zeta function explained

In mathematics, the Airy zeta function, studied by, is a function analogous to the Riemann zeta function and related to the zeros of the Airy function.

Definition

The Airy function

Ai(x)=

	1
	\pi

\cos\left(\tfrac13t³+xt\right)dt,

is positive for positive x, but oscillates for negative values of x. The Airy zeros are the values

\{a_i\}

at which

Ai(a_i)=0

, ordered by increasing magnitude:

|a_1|<|a_{2|< …}

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

\zeta_Ai

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.

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_Ai

5/3

4(	23)

\Gamma

4\pi²

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_Ai(1)=-

\Gamma(	23)

\Gamma(	43)\sqrt[3]{9

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Airy zeta function".