Airy Zeta Function

Author

Eric W. Weisstein
July 6, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/AiryZetaFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/AiryZetaFunction.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Utilities`Simplify`

CRatio[z_]:=AiryBi[z]/AiryAi[z]
T[n_,z_]:=((Derivative[n][CRatio][z])/((3^(2/3)) ((Gamma[(1/3)])^2)))-Sum[Binomial[n,j]Derivative[n-j][CRatio][z]Derivative[j-1][AiryAi[#]^2&][z],{j,n}]+Derivative[n-1][AiryAi[#]AiryBi[#]&][z]
AiryZeta[n_]:=π T[n-1,0]/Gamma[n]

Function

Leading Constant

Integrate[AiryAi[u]^2,{u,0,∞}]

Constant X

(1/(2π AiryAi[0]AiryBi[0]))

GammaSimplify[%]

Polynomial

Solve[X==((3^(5/6))/(2π))((GammaSimplify[Gamma[2/3]])^2)/.Gamma[1/3]:>g,g]

(tri=Table[
GammaSimplify[AiryZeta[n]]/.Gamma[1/3]:>(Sqrt[2π])/(3^(1/12))/(Sqrt[X]),
{n,2,20}]//Simplify)//ColumnForm

Times@@FactorTermsList[#]&/@tri//ColumnForm//TraditionalForm

DeleteCases[CoefficientList[FactorTermsList[#],X][[2]],0]&/@tri

Flatten[%]

1/FactorTermsList[#][[1]]&/@tri

Plot

<<Utilities`Plot`

IntegerListPlot[AiryZeta[#]&/@Range[2,10],2,PlotRange->All,AxesLabel->TraditionalForm/@{n,Z[n]}]

Values

AiryZeta[#]&/@Range[2,10]//N

Computation

<<Utilities`Plot`

a=PlotRoots[AiryAi[z],{z,-100,2},PlotPoints->500];

Plus@@(1/a^#)&/@Range[2,10]