mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Airy zeta function, studied by , is a function analogous to the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

and related to the zeros of the Airy function.

Definition

The Airy function :

\mathrm(x) = \frac \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,

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

\_^\infty

at which

\text(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_(s)=\sum_^ \frac.

This series converges when the real part of ''s'' is greater than 3/2, and may be extended by

analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...

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_(2)=\sum_^ \frac=\frac,

where

\Gamma

is the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

, a continuous variant of the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...

. 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_(1)=-\frac.

References

External links

*{{MathWorld, title=Airy Zeta Function, urlname=AiryZetaFunction Zeta and L-functions