Tau Function

Author

Eric W. Weisstein
December 18, 2015

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

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

©2015 Wolfram Research, Inc. except for portions noted otherwise

Plot

<<Utilities`Plot`

IntegerListPlot[Table[RamanujanTau[n],{n,100}],
PlotJoined->True,PlotStyle->Red,PlotRange->All,AxesLabel->TraditionalForm/@{n,τ[n]}]

Sums

(∑_n^∞)(RamanujanTau[n]) (x^n)==x ((((QPochhammer2[x, x])))^24)

V7

Sum[RamanujanTau[n](x^n),{n,∞}]

Sum[RamanujanTau[n](x^n),{n,10}]

x Product[(((1-(x^n)))^24),{n,∞}]

FullSimplify[%]

V8

$Version

Sum[RamanujanTau[n](x^n),{n,0,∞}]

Sum[RamanujanTau[n](x^n),{n,1,∞}]

Sum[RamanujanTau[n](x^n),{n,∞}]

(∑_n^∞)(((-1))^n) (RamanujanTau[n]) (x^n)==-x ((((QPochhammer2[-x, -x])))^24)

Sum[(((-1))^n)RamanujanTau[n](x^n),{n,∞}]

Sum[(((-1))^n)RamanujanTau[n](x^n),{n,10}]

Series[(-x)((QPochhammer[-x,-x])^24),{x,0,10}]

Generating Function

(∑_(n=0)^10)(RamanujanTau[n]) (x^n)==x (∏_n^∞)(((1-(x^n)))^24)

V6

Table[RamanujanTau[n],{n,10}]

Series[x Product[(1-x^n)^24,{n,100}],{x,0,10}]

x Product[(1-x^n)^24,{n,∞}]

SeriesCoefficient[x ((QPochhammer[x,x])^24),{x,0,n}]

Sum[RamanujanTau[n](x^n),{n,0,∞}]

Sum[RamanujanTau[n](x^n),{n,0,10}]

GeneratingFunction[RamanujanTau[n],n,x]

Series[x ((QPochhammer[x,x])^24),{x,0,10}]

V8

SeriesCoefficient[x ((QPochhammer[x])^24),{x,0,n}]

GeneratingFunction[RamanujanTau[n],n,x]

Ewell

ExactPowerOf2[n_]:=Log[2,EvenPart[n]]
b[n_]:=ExactPowerOf2[n]

Equation 1

tau[i_]:=Module[{n=(i-2)/4,k,j},
-3 Sum[2^(3b[2k])DivisorSigma[3,OddPart[2k]]Sum[(-1)^j SumOfSquaresR[8,4n+2-2k-j]SumOfSquaresR[8,j],{j,0,4n+2-2k}],{k,2n+1}]
]

-3 (∑_(k=1)^(2 n+1))(2^(3 b(2 k))) (Subscript[σ, 3])(Od(2 k)) (∑_(j=0)^(-2 k+4 n+2))(((-1))^j) (Subscript[r, 8])(-j-2 k+4 n+2) (Subscript[r, 8])(j)

Table[{tau[#],RamanujanTau[#]}&[4n+2],{n,10}]

Equation 2

tau[n_]:=Sum[2^(3b[2k])DivisorSigma[3,OddPart[2k]]Sum[(-1)^j SumOfSquaresR[8,2n+1-2k-j]SumOfSquaresR[8,j],{j,0,2n+1-2k}],{k,n}]

Table[tau[n],{n,20}]

Equation 3

tau[n_]:=Module[{m=n/4},
-2^11 RamanujanTau[m]-3 Sum[2^(3b[2k]) DivisorSigma[3,OddPart[2k]] Sum[(-1)^j SumOfSquaresR[8,4m-2k-j]SumOfSquaresR[8,j],{j,0,4m-2k}],{k,2m}]]

Table[{tau[#],RamanujanTau[#]}&[4n],{n,10}]

Equation 4

tau[i_]:=Module[{n=(i-1)/2,k,j},
Sum[2^(3(b[2k]-1)) DivisorSigma[3,OddPart[2k]] Sum[(-1)^j SumOfSquaresR[8,4n+2-2k-j]SumOfSquaresR[8,j],{j,0,4n+2-2k}],{k,2n+1}]]

Table[{tau[#],RamanujanTau[#]}&[2n+1],{n,10}]

Residues

Union[Table[RamanujanTau[Mod[2n,2]],{n,1000}]]

Union[Table[RamanujanTau[Mod[3 n,3]],{n,1000}]]

Union[Table[RamanujanTau[Mod[5n,5]],{n,1000}]]

Union[Table[RamanujanTau[Mod[7n,7]],{n,10000}]]

Ewell

Table[Mod[RamanujanTau[4n]-RamanujanTau[n],3],{n,20}]

Nonquadatic Residue

Nonquadatic Residues (mod 7)

Complement[Range[6],Union[Table[Mod[i^2,7],{i,0,6}]]]

Nonquadratic Residues (mod 23)

nonq23=Complement[Range[22],Union[Table[Mod[i^2,23],{i,0,22}]]]

Union[Flatten[Table[Mod[RamanujanTau[23n+#],23]&/@nonq23,{n,100}]]
]//Timing