Xi-Function

Author

Eric W. Weisstein
December 28, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Xi-Function.nb.

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

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

Definition

XiFunction[x]

FullSimplify[Subtract@@{XiFunction[x],(x/2)!(x-1)(π^(-x/2))Zeta[x]}]

Special Values

Table[{n,Limit[XiFunction[x],x->n]},{n,0,5}]//TableForm

Real

FullSimplify[Im[XiFunction[t]],t∈Reals]

FullSimplify[Im[XiFunction[1/2+I t]],t∈Reals]

Plot[ Im[(π^(-(1/4)-((I t)/2))) Gamma[(1/4) (1+2 I t)] Zeta[(1/2)+I t]],{t,-5,5},PlotStyle->Red]

Integrals

(1/2)-(1/2) (s (1-s)) (∫_1^∞)((((x^(s/2))+(x^((1-s)/2))) ((EllipticTheta[3, 0, (e^(-π s))])-1))/(2 x))dx==ξ(s)

Psi[x_]:=(1/2) (-1+EllipticTheta[3,0,(E^(-π x))])

Table[With[{s=Random[Real,{0,1},20]},
{
s,
(1/2)-((s(1-s))/2)NIntegrate[Psi[s]((x^(s/2))+(x^((1-s)/2)))(1/x),{x,1,1000}],
XiFunction[s]
}],{10}]

V5

(1/2)-((s(1-s))/2)Integrate[Psi[s]((x^(s/2))+(x^((1-s)/2)))(1/x),{x,1,∞}]

With[{s=.5},Plot[Psi[s]((x^(s/2))+(x^((1-s)/2)))(1/x),{x,1,1000},PlotStyle->Red]]

With[{s=.5},(1/2)-((s(1-s))/2)NIntegrate[Psi[s]((x^(s/2))+(x^((1-s)/2)))(1/x),{x,1,10,20}]]

XiFunction[.5]

With[{s=0},(1/2)-((s(1-s))/2)Integrate[Psi[s]((x^(s/2))+(x^((1-s)/2)))(1/x),{x,1,∞}]]

With[{s=1},(1/2)-((s(1-s))/2)Integrate[Psi[s]((x^(s/2))+(x^((1-s)/2)))(1/x),{x,1,∞}]]

With[{s=2},(1/2)-((s(1-s))/2)Integrate[Psi[s]((x^(s/2))+(x^((1-s)/2)))(1/x),{x,1,∞}]]

Plot[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,10},PlotStyle->Red,PlotRange->All]

v5.2

(1/2)-(π/6)+NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,10},WorkingPrecision->50]

(π/6)-(1/2)//N

NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,10},WorkingPrecision->50]

NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,∞},WorkingPrecision->50]

NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,10,∞},WorkingPrecision->30]

(1/2)+NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,2,∞},Method->QuasiMonteCarlo,WorkingPrecision->10,MaxPoints->10^5]

v6

(1/2)+NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,2,∞},WorkingPrecision->30]

NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,2,∞},Method->AdaptiveQuasiMonteCarlo,WorkingPrecision->20,MaxPoints->10^5]

NIntegrate[((((1/(Sqrt[x]))+x) (-1+EllipticTheta[3,0,(E^(-2 π))]))/(2 x)),{x,1,10},WorkingPrecision->10]

Sums

(∑_n^∞)exp(π (-(n^2)) x)==(((EllipticTheta[3, 0, (e^(-π x))])-1)/2)

Sum[Exp[-(n^2)π x],{n,∞}]

Products

(1/2) (∏_(k=-∞)^∞)If[k==0,1,1-(s/(ZetaZero[k]))]==(1/2) (π^(-(s/2))) (s-1) s (Zeta[s]) (Gamma[(s/2)])

(1/2)Product[If[k==0,1,1-(s/(ZetaZero[k]))],{k,-∞,∞}]

With[{s=.1},(1/2)Product[If[k==0,1,1-(s/(ZetaZero[k]))],{k,-20,20}]]//Chop

XiFunction[s]

XiFunction[.1]

With[{s=7.8},(1/2)Product[If[k==0,1,1-(s/(ZetaZero[k]))],{k,-500,500}]]//Chop//Timing

With[{s=7.8},(1/2)Product[If[k==0,1,1-(s/(ZetaZero[k]))],{k,-2000,2000}]]//Chop//Timing

XiFunction[7.8]

XiFunction[s]

Series

V5

<<DiscreteMath`RSolve`

SeriesTerm[XiFunction[x],{x,1/2,n}]

l=CoefficientList[Series[XiFunction[x],{x,1/2,3}]/.x->z+1/2,z];

N[l,10]

FullSimplify[l[[1]]]

FullSimplify[l[[2]]]

Simplify[l[[3]]]

TraditionalForm[%40]

Simplify[l[[4]]]

psi[x_]:=(1/2) (-1+EllipticTheta[3,0,(E^(-π x))])

Integrate[D[(x^(3/2))psi'[x],x](x^(-1/4))((((Log[x]/2))^(2n))/((2n)!)),{x,1,∞}]

i[n_]:=4NIntegrate[D[(x^(3/2))psi'[x],x](x^(-1/4))((((Log[x]/2))^(2n))/((2n)!)),{x,1,∞},WorkingPrecision->20]

Table[i[n],{n,0,2}]

V7

XiFunction[x]

TimeConstrained[SeriesCoefficient[XiFunction[x],{x,0,n}],300]//Timing

TimeConstrained[SeriesCoefficient[XiFunction[x],{x,1/2,n}],300]//Timing

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Axis

Plot[XiFunction[x],{x,-5,5},
AxesLabel->TraditionalForm/@{x,XiFunction[x]},
PlotStyle->Red]

Complex Plane

ComplexPlot3D[XiFunction[z],{z,-5,5}]//Timing

ComplexContourPlot[XiFunction[z],{z,-5,5}]//Timing

Xi Roots

<<Utilities`Plot`

PlotRoots[Re[XiFunction[1/2+I t]],{t,0,50},PlotRange->10^-4{-1,1},PlotStyle->Blue,AxesLabel->TraditionalForm/@{t,HoldForm[XiFunction[1/2+I t]]}]

Comparison of Xi and Zeta Roots

xi=PlotRoots[Re[XiFunction[1/2+I x]],{x,-50,50},PlotStyle->Blue,AxesLabel->TraditionalForm/@{x,HoldForm[XiFunction[1/2+I x]]}]

re=PlotRoots[Re[Zeta[1/2+I x]],{x,-50,50},PlotPoints->100,PlotStyle->Blue,AxesLabel->TraditionalForm/@{x,HoldForm[Re[Zeta[1/2+I x]]]}]

im=PlotRoots[Im[Zeta[1/2+I x]],{x,-50,50},PlotPoints->100,PlotStyle->Blue,AxesLabel->TraditionalForm/@{x,HoldForm[Im[Zeta[1/2+I x]]]}]

zeta=Intersection[re,im,SameTest->(Abs[#1-#2]<10^-4&)]

Complement[zeta,xi,SameTest->(Abs[#1-#2]<10^-4&)]