Dirichlet Generating Function

Author

Eric W. Weisstein
May 18, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Examples

Möbius function

Sum[MoebiusMu[n]/(n^s),{n,∞}]

DirichletTransform[MoebiusMu[n],n,s]

1

Sum[1/n^s,{n,∞}]

DirichletTransform[1,n,s]

Plot[Zeta[s],{s,0,5}]

Plot[Zeta[s]-Evaluate[Sum[1/n^s,{n,10^4}]],{s,0,5}]

DivisorSigma[0,n]

Sum[DivisorSigma[0,n]/n^s,{n,∞}]

DirichletTransform[DivisorSigma[0,n],n,s]

DivisorSigma[k,n]

Sum[DivisorSigma[k,n]/n^s,{n,∞}]

DirichletTransform[DivisorSigma[k,n],n,s]

Table[DivisorSigma[0,n],{n,10}]

PrimeNu

Sum[PrimeNu[n]/n^s,{n,∞}]

DirichletTransform[PrimeNu[n],n,s]

((1-(((-1))^n))/(2 (n^s)))

Sum[(1/2) (1-(((-1))^n))/n^s,{n,∞}]//Simplify

DirichletTransform[(1/2) (1-(((-1))^n)),n,s]//FullSimplify

Table[-((-1)^n-1)/2,{n,10}]

lsb[n]

Sum[((Mod[n,2])/(n^s)),{n,∞}]//FullSimplify

DirichletTransform[Mod[n,2],n,s]//FullSimplify

Plot[(1-(2^(-s))) Zeta[s],{s,1,5}]

Ruler function values

A001511

Sum[((IntegerExponent[2*n,2])/(n^s)),{n,∞}]

DirichletTransform[IntegerExponent[2*n,2], n,s]

Perrin sequence

A001608

Sum[((RootSum[-1-#1+#1^3&,#1^n&])/(n^s)),{n,∞}]

DirichletTransform[RootSum[-1-#1+#1^3&,#1^n&],n,s]

sum=Sum[((RootSum[-1-#1+#1^3&,#1^n&])/(n^s)),{n,200}];

Plot[Zeta[s-1]-sum,{s,0,5},PlotStyle->Thick]

Dedekind Psi

DirichletTransform[n Exp[DivisorSum[n,Log[1+1/#]&,PrimeQ]],n,s]

Pancake-cutting

A000124

Sum[((Binomial[n+1,2]+1)/(n^s)),{n,∞}]

FullSimplify[%,s>0]

DirichletTransform[Binomial[n+1,2]+1,n,s]

Divisions by Chords

A000127

Sum[(((1/24) (24-18 n+23 (n^2)-6 (n^3)+(n^4)))/(n^s)),{n,∞}]//FullSimplify

(1/24) (Zeta[-4+s]-6 Zeta[-3+s]+23 Zeta[-2+s]-18 Zeta[-1+s])+Zeta[s]/.s->.5

Zeta[s-1]/.s->.5

DirichletTransform[(1/24) (24-18 n+23 (n^2)-6 (n^3)+(n^4)),n,s]//FullSimplify

Plot[(1/24) (Zeta[-4+s]-6 Zeta[-3+s]+23 Zeta[-2+s]-18 Zeta[-1+s])+Zeta[s],{s,1,5}]

Cubefree Part

A050985

zeta(3s)*zeta(s-1)/zeta(3s-3)

Plot[((Zeta[3s]Zeta[s-1])/(Zeta[3s-3])),{s,0,5}]

fn=Total[MapIndexed[#/#2[[1]]^s&,{1,2,3,4,5,6,7,1,9,10,11,12,13,14,15,2,17,18,19,20,21,22,23,3,25,26,1,28,29,30,31,4,33,34,35,36,37,38,39,5,41,42,43,44,45,46,47,6,49,50,51,52,53,2,55,7,57,58,59,60,61,62,63,1,65,66,67,68,69,70,71,9,73,74,75}]]

Plot[((Zeta[3s]Zeta[s-1])/(Zeta[3s-3])) -fn,{s,0,5}]

Largest odd divisor

A000265

Sum[(n/2^IntegerExponent[n,2])/n^s,{n,Infinity}]

Sum[(n/2^IntegerExponent[n,2])/n^s,{n,Infinity},Assumptions->s>1]//Timing

Sum[(n/2^IntegerExponent[n,2])/n^s,{n,Infinity},Assumptions->s>2]//Timing

SumConvergence[(n/2^IntegerExponent[n,2])/n^s,n]

DirichletTransform[(n/2^IntegerExponent[n,2]),n,s]

sum=Sum[(n/2^IntegerExponent[n,2])/n^s,{n,1000}];

Plot[Zeta[s-1]*(2^s-2)/(2^s-1)-sum,{s,0,5},PlotStyle->Thick]

Rhombic dodecahedral

A005917

DirichletTransform[(n+1)^4-n^4,n,s]