Mangoldt Function

Author

Eric W. Weisstein
August 5, 2015

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

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

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

Code

MyMangoldtLambda[1]:=0
MyMangoldtLambda[n_Integer?Positive]:=Module[{f=FactorInteger[n]},
If[Length[f]==1,Log[f[[1,1]]],0]
]

Notation

MangoldtLambda[n]

Enumeration

MangoldtLambda[Range[10]]

Exp/@%

Identities

Table[LCM@@Range[n]/LCM@@Range[n-1],{n,2,10}]

Sums

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

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

DivisorSum[n,(MangoldtLambda[#1])&]==ln(n)

DivisorSum[n,MangoldtLambda[#]&]

DivisorSum[n,(MoebiusMu[#1]) ln(#1)&]==-(MangoldtLambda[n])

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

DivisorSum[n,(MangoldtLambda[#1]) (MoebiusMu[(n/(#1))])&]==-(MoebiusMu[n]) ln(n)

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

Subtract@@@Table[{DivisorSum[n,MoebiusMu[(n/(#))]MangoldtLambda[#]&],-MoebiusMu[n]Log[n]},{n,1000}]//Simplify//Union

(∑_n^∞)((MangoldtLambda[n])/(n^s))==-((ζ'(s))/(Zeta[s]))

Sum[((MangoldtLambda[n])/(n^s)),{n,∞}]//InputForm

Subtract@@@Table[{Sum[((MangoldtLambda[n])/(n^s)),{n,5 10^4}],-((Zeta'[s])/(Zeta[s]))}/.{s->RandomReal[{1,10}]+RandomReal[{-10,10}]I},{10}]//Chop//Union

Plots

Show[GraphicsArray[{
IntegerListPlot[Table[MangoldtLambda[n],{n,100}],
PlotStyle->Red,
DisplayFunction->Identity,AxesLabel->TraditionalForm/@{n,Λ[n]}],
IntegerListPlot[Exp/@Table[MangoldtLambda[n],{n,100}],
PlotStyle->Red,
DisplayFunction->Identity,
PlotRange->All,AxesLabel->TraditionalForm/@{n,Exp[Λ[n]]}]
}],GraphicsSpacing->-.07,ImageSize->500]

Summatory

Sum[MangoldtLambda[n],{n,0,x}]

Sum[MangoldtLambda[n],{n,x}]

SummatoryMangoldtPsi[x_]:=Sum[MangoldtLambda[n],{n,x}]

Identitities

(LogGamma[(Floor[x])+1])==(∑_(k=1)^x)SummatoryMangoldtPsi((x/k))

Plot[Sum[SummatoryMangoldtPsi[x/k],{k,1,100}]-LogGamma[Floor[x]+1],{x,0,18}]