Möbius Function

Author

Eric W. Weisstein
December 21, 2009

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

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

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

Values

MoebiusMu/@Range[20]

{#,Flatten[Position[MoebiusMu/@Range[30],#]]}&/@Range[-1,1]//ColumnForm

Plot

<<Utilities`Plot`

IntegerListPlot[MoebiusMu[Range[100]],PlotStyle->Red,
AxesLabel->TraditionalForm/@{n,MoebiusMu[n]}]

Show[PixelPerfect[{MoebiusMu/@Range[100]},BlockSize->3,
ColorFunction->(Switch[#,-1,Red,0,Black,1,Blue]&),
ColorFunctionScaling->False]]

ListDensityPlot[Partition[MoebiusMu/@Range[0,100^2],100],
ColorFunction->(Switch[#,-1,Red,0,Black,1,Blue]&),
ColorFunctionScaling->False,Mesh->False,FrameLabel->TraditionalForm/@{Mod[n,100],Floor[n/100]}]

l=Flatten[Position[MoebiusMu/@Range[200],#]]&/@Range[-1,1];
p=MapIndexed[Flatten[{#2,#1}]&,#]&/@l;

<<Graphics`Colors`

Show[Graphics[{
PointSize[.02],
Transpose[{
{Red,Black,Blue},{Line[#],Point/@#}&/@p
}]
}
],Axes->True,PlotRange->{{-1,50},{-5,160}}]

Sums

(∑_n^∞)(((MoebiusMu[n]) (x^n))/(1-(x^n)))==x

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

(∑_k^∞)((MoebiusMu[k])/k)==0

Sum[MoebiusMu[k]/k,{k,Infinity}]

(∑_k^∞)((ln(k) (MoebiusMu[k]))/k)==-1

V6

$Version

Sum[MoebiusMu[k]/k Log[k],{k,Infinity}]//InputForm

V7

$Version

Sum[MoebiusMu[k]/k Log[k],{k,Infinity}]//InputForm

NSum[MoebiusMu[k]/k Log[k],{k,Infinity}]

Sum[MoebiusMu[k]/k Log[k],{k,10^6}]//N

V8

Sum[MoebiusMu[k]/k Log[k],{k,Infinity}]

(∑_(n=1)^∞)((μ(n))/(n^s))

Sum[MoebiusMu[n]/n^s,{n,∞}]

(∑_(n=1)^∞)((|μ(n)|)/(n^2))

Plus@@N[Table[Abs[MoebiusMu[n]]/n^2,{n,10^7}],20]//Timing

N[15/Pi^2,20]

V6

V7

Sum[((Abs[MoebiusMu[n]])/(n^2)),{n,∞}]//InputForm

(∑_(n=1)^∞)(Subscript[χ, {n| μ(n)=1} ])~ 0.304016 n

l=1-Unitize[MoebiusMu[Range[10^7]]-1];//Timing

f=MapIndexed[{#2[[1]],#/#2[[1]]}&,Accumulate[N[Take[l,10^6]]]][[1;;-1;;1000]];//Timing

ListPlot[f]

With[{n=5*^6},Total[Take[l,n]//N]/n]

With[{n=7*^6},Total[Take[l,n]//N]/n]

N[Total[l]]/Length[l]

3./Pi^2

(∑_(n=1)^∞)((Subscript[χ, {n| μ(n)=1}])/(n^2))

N[21/(2Pi^2),20]

Plus@@N[Table[KroneckerDelta[1,MoebiusMu[n]]/n^2,{n,10^6}],20]//Timing

Plus@@N[Table[KroneckerDelta[1,MoebiusMu[n]]/n^2,{n,10^7}],20]//Timing

V7

Sum[((Boole[MoebiusMu[n]==-1])/(n^2)),{n,∞}]//InputForm

Sum[((Boole[MoebiusMu[n]==-1])/(n^2)),{n,10^5}]//N

9/(2(π^2))//N

(∑_(n=1)^∞)((Subscript[χ, {n| μ(n)=-1}])/(n^2))

N[9/(2Pi^2),20]

Plus@@N[Table[KroneckerDelta[-1,MoebiusMu[n]]/n^2,{n,10^6}],20]//Timing

Plus@@N[Table[KroneckerDelta[-1,MoebiusMu[n]]/n^2,{n,10^7}],20]//Timing

V7

Sum[((Boole[MoebiusMu[n]==1])/(n^2)),{n,∞}]//InputForm

Sum[((Boole[MoebiusMu[n]==1])/(n^2)),{n,10^5}]//N

21/(2(π^2))//N

DivisorSum[n,(MoebiusMu[#1])&]==(Subscript[δ, 1,n])

DivisorSum[n,MoebiusMu]

DivisorSum[n,MoebiusMu[#]&]

DivisorSum[n,(MoebiusMu[#1]) (DivisorSigma[0, (n/(#1))])&]==1

DivisorSum[n,MoebiusMu[#]DivisorSigma[0,(n/(#))]&]

Table[DivisorSum[n,MoebiusMu[#]DivisorSigma[0,(n/(#))]&],{n,100}]//Union

DivisorSum[n,#1 (MoebiusMu[(n/(#1))])&]==(EulerPhi[n])

DivisorSum[n,# MoebiusMu[(n/(#))]&]

DivisorSum[n,|(MoebiusMu[#1])|&]==(2^(PrimeNu[n]))

DivisorSum[n,Abs[MoebiusMu[#]]&]

Subtract@@@Table[{DivisorSum[n,Abs[MoebiusMu[#]]&],2^PrimeNu[n]},{n,1000}]//Union

Products

(∏_n^∞)(((1-(x^n)))^((MoebiusMu[n])/n))==exp(-x)

f[x_]:=NProduct[(1-x^n)^(MoebiusMu[n]/n),{n,Infinity}]

Plot[f[x]-Exp[-x],{x,0,.9},PlotRange->All]

V6

Product[(1-x^n)^(MoebiusMu[n]/n),{n,∞}]

V7

Product[(1-x^n)^(MoebiusMu[n]/n),{n,∞}]//InputForm