Tau Dirichlet Series

Author

Eric W. Weisstein
January 11, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheoreticFunctions/TauDirichletSeries.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Plots

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

Show[GraphicsArray[Plot[RamanujanTauL[s],Evaluate[{s,Sequence@@#}],
PlotStyle->Red,
DisplayFunction->Identity,
AxesLabel->TraditionalForm/@{s,RamanujanTauL[s]}
]&/@{{-5,1},{0,10}}
]]

ComplexPlot3D[RamanujanTauL[s],{s,-5-I,1+I}]//Timing

ComplexContourPlot[RamanujanTauL[s],{s,-5-I,1+I}]//Timing

Products

(∏_n^∞)(1/(-(RamanujanTau[(Prime[n])]) ((((Prime[n])))^(-s))+((((Prime[n])))^(11-2 s))+1))==RamanujanTauL(s)

Product[(1/(1-RamanujanTau[Prime[n]]((Prime[n])^(-s))+((Prime[n])^(11-2s)))),{n,∞}]

Subtract@@@Table[{Product[(1/(1-RamanujanTau[Prime[n]]((Prime[n])^(-s))+((Prime[n])^(11-2s)))),{n,2000}],RamanujanTauL[s]}/.s->RandomReal[{7,20},WorkingPrecision->70]+RandomReal[{-5,5},WorkingPrecision->70]I,{20}]//Chop//Union

Sums

(∑_n^∞)((RamanujanTau[n])/(n^s))==RamanujanTauL(s)

V7.0.1

Sum[((RamanujanTau[n])/(n^s)),{n,∞}]

Functional Equations

((RamanujanTauL(s) (Gamma[s]))/(((2 π))^s))==((RamanujanTauL(12-s) (Gamma[12-s]))/(((2 π))^(12-s)))

FullSimplify[((RamanujanTauL[s]Gamma[s])/(((2π))^s))-((RamanujanTauL[12-s]Gamma[12-s])/(((2π))^(12-s)))]

Table[-(((2 π))^(-12+s)) Gamma[12-s] RamanujanTauL[12-s]+(((2 π))^(-s)) Gamma[s] RamanujanTauL[s]/.s->RandomComplexDisk[],{100}]//Chop//Union

Plot[RamanujanTauL[s]Gamma[s]/(2Pi)^s-RamanujanTauL[12-s]Gamma[12-s]/(2Pi)^(12-s),{s,0,12},PlotStyle->Red]

Identities

RamanujanTauL(6+i t)==(e^(-i RamanujanTauTheta(t))) RamanujanTauZ(t)

FunctionExpand[RamanujanTauL[6+I t],t∈Reals]

FullSimplify[RamanujanTauZ[t]Exp[-I RamanujanTauTheta[t]]]

FullSimplify[RamanujanTauZ[t]Exp[-I RamanujanTauTheta[t]],t∈Reals]

Subtract@@@Table[{(E^(-I RamanujanTauTheta[t])) RamanujanTauZ[t],RamanujanTauL[6+I t]}/.t->RandomReal[{-2,2}],{50}]//Chop//Union

RamanujanTauZ(t)==(((2 π))^(-i t)) RamanujanTauL(6+i t) (Sqrt[((sinh(π t))/(π t ((t^2)+1) ((t^2)+4) ((t^2)+9) ((t^2)+16) ((t^2)+25)))]) (Gamma[i t+6])

FunctionExpand[RamanujanTauZ[t]]

FullSimplify[Gamma[6+I t]RamanujanTauL[6+I t](((2π))^(-I t))(Sqrt[((Sinh[π t])/(π t(1+(t^2))(4+(t^2))(9+(t^2))(16+(t^2))(25+(t^2))))])]

FullSimplify[Gamma[6+I t]RamanujanTauL[6+I t](((2π))^(-I t))(Sqrt[((Sinh[π t])/(π t(1+(t^2))(4+(t^2))(9+(t^2))(16+(t^2))(25+(t^2))))])-RamanujanTauZ[t]]

RamanujanTauTheta(t)==-t log(2 π)-(1/2) i log(((Gamma[i t+6])/(Gamma[6-i t])))

FunctionExpand[RamanujanTauTheta[t]]

FullSimplify[-(1/2)I Log[((Gamma[6+I t])/(Gamma[6-I t]))]-t Log[2π]]