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.
Contents
Definition
The Airy function
A
i
(
x
)
=
1
π
∫
0
∞
cos
(
1
3
t
3
+
x
t
)
d
t
,
{\displaystyle \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
∞
{\displaystyle \{a_{i}\}_{i=1}^{\infty }}
at which
Ai
(
a
i
)
=
0
{\displaystyle {\text{Ai}}(a_{i})=0}
, ordered by increasing magnitude:
|
a
1
|
<
|
a
2
|
<
⋯
{\displaystyle |a_{1}|<|a_{2}|<\cdots }
.
The Airy zeta function is the function defined from this sequence of zeros by the series
ζ
A
i
(
s
)
=
∑
i
=
1
∞
1
|
a
i
|
s
.
{\displaystyle \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
ζ
(
2
)
=
π
2
/
6
{\displaystyle \zeta (2)=\pi ^{2}/6}
is the solution to the Basel problem,
the Airy zeta function may be exactly evaluated at s = 2:
ζ
A
i
(
2
)
=
∑
i
=
1
∞
1
a
i
2
=
3
5
/
3