Divisor Function

Author

Eric W. Weisstein
September 17, 2019

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

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

©2019 Wolfram Research, Inc. except for portions noted otherwise

Plot

GraphicsGrid[Partition[Table[ListStepPlot[Table[DivisorSigma[k,n],{n,100}],PlotStyle->{AbsoluteThickness[.5],Red},LabelStyle->FontFamily->"Times",AxesLabel->{n,DivisorSigma[k,n]}],{k,0,3}],2]]

Generating Functions

(DivisorSigma[0, n])

Sum[x^k/(1-x^k),{k,∞}]

Series[((QPolyGamma[1,x]+Log[1-x])/(Log[x])),{x,0,10}]

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

Generating Function

Sum[DivisorSigma[0,n]x^n,{n,∞}]

GeneratingFunction[DivisorSigma[0,n],n,z]

ExponentialGeneratingFunction

Sum[DivisorSigma[0,n]((x^n)/(n!)),{n,∞}]

ExponentialGeneratingFunction[DivisorSigma[0,n],n,z]

DirichletGeneratingFunction

Sum[((DivisorSigma[0,n])/(n^s)),{n,∞}]

Lambert Series

Sum[((DivisorSigma[0,n](x^n))/(1-(x^n))),{n,∞}]

Sums

(∑_k^n)((Floor[(n/k)])-(Floor[((n-1)/k)]))==(DivisorSigma[0, n])

Sum[Floor[(n/k)]-Floor[((n-1)/k)],{k,n}]

Table[{Sum[Floor[(n/k)]-Floor[((n-1)/k)],{k,n}],DivisorSigma[0,n]},{n,10}]

Series

(DivisorSigma[1, n])

V6

Series[DivisorSigma[1,n],{n,0,3}]

Refine[n Table[(1/(i^2))Cos[π n (j/i)],{i,0,4},{j,i}],n∈Integers]

Sum[((((-1))^n)/(k^2)),{k,∞}]

Table[Cos[n π/2],{n,5}]

n*sum(i=1,infinity)sum(j=1,i)1/i^2*Cos[2Pi*n*j/i] =
n(1/6*Pi^2(1+1/2^2*(-1)^n)) + 2Cos[Pi*n*2/3]/3^2 + 2Cos[Pi*n*1/2]/4^2 + 2(Cos[Pi*n*2/5]+Cos[Pi*n*4/5])/5^2...

V7

Series[DivisorSigma[1,n],{n,0,3}]

V12

Series[DivisorSigma[1,n],{n,0,3}]

Sum[1/i^2*Cos[2Pi*n*j/i],{j,i},{i,∞}]

Sum[1/i^2*Cos[2Pi*n*j/i],{i,4},{j,i}]

Sums

(∑_n^∞)(((DivisorSigma[0, n])^2)/(n^s))==((((ζ(s)))^4)/(ζ(2 s)))

V6

Sum[(((DivisorSigma[0,n])^2)/(n^s)),{n,∞}]

V7

Sum[(((DivisorSigma[0,n])^2)/(n^s)),{n,∞}]

(∑_n^∞)((DivisorSigma[0, n])/(n^s))==(((ζ(s)))^2)

Sum[((DivisorSigma[0,n])/(n^s)),{n,∞}]

DivisorSum[n,((#1)^k)&]==(DivisorSigma[k, n])

DivisorSum[n,#^k&]

(∑_n^∞)((DivisorSigma[1, n])/(n^s))==ζ(s-1) ζ(s)

Sum[((DivisorSigma[1,n])/(n^s)),{n,∞}]

(∑_n^∞)((DivisorSigma[k, n])/(n^s))==ζ(s) ζ(s-k)

Sum[((DivisorSigma[k,n])/(n^s)),{n,∞}]

(∑_n^∞)(((Subscript[σ, a])(n) (Subscript[σ, b])(n))/(n^s))==((ζ(s) ζ(s-a) ζ(s-b) ζ(-a-b+s))/(ζ(-a-b+2 s)))

Ramanujan (Hardy 1999, p. 59)

V6

Sum[(DivisorSigma[a, n]*DivisorSigma[b, n])/n^s, {n, Infinity}]//InputForm

f[{a_,b_},s_]:={Sum[DivisorSigma[a,n]DivisorSigma[b,n]/n^s,{n,Infinity}],Zeta[s]Zeta[s-a]Zeta[s-b]Zeta[s-a-b]/Zeta[2s-a-b]}

f[{3,4},10]

N[%,20]

NSum[((DivisorSigma[3,n] DivisorSigma[4,n])/(n^10)),{n,∞}]

NSum[((DivisorSigma[3,n] DivisorSigma[4,n])/(n^10)),{n,∞},NSumTerms->50,
NSumExtraTerms->100,WorkingPrecision->100]

N[((DivisorSigma[3,n] DivisorSigma[4,n])/(n^10))/.n->100]

V7

Sum[(DivisorSigma[a, n]DivisorSigma[b, n])/n^s, {n, Infinity}]

1+240 (∑_n^∞)(DivisorSigma[3, n]) (q^n)==(1/2) ((((Subscript[ϑ, 2])(0,(Sqrt[q])))^8)+(((Subscript[ϑ, 3])(0,(Sqrt[q])))^8)+(((Subscript[ϑ, 4])(0,(Sqrt[q])))^8))

(1+240 Sum[DivisorSigma[3,n]q^n,{n,∞}])

120 (∑_k^(n-1))(DivisorSigma[3, k]) (DivisorSigma[3, n-k])==(DivisorSigma[7, n])-(DivisorSigma[3, n])

120Sum[DivisorSigma[3,n-k]DivisorSigma[3,k],{k,n-1}]//InputForm

Subtract@@@Table[{120Sum[DivisorSigma[3,n-k]DivisorSigma[3,k],{k,n-1}],DivisorSigma[7,n]-DivisorSigma[3,n]},{n,1000}]//Union

Others

Inequalities

Gronwall's theorem

With[{c=Exp[EulerGamma],maxx=10^4},ListPlot[Table[((DivisorSigma[1,n])/(n Log[Log[n]])),{n,maxx}],AxesLabel->{n,((DivisorSigma[1,n])/(Row[{n," ","(",ln,"  ",ln," ",n,")"}]))},LabelStyle->FontFamily->"Times",PlotRangeClipping->None,PlotRange->{Automatic,{.4,c+.1}},Epilog->{Red,Text[Style[c,FontFamily->"Times"],{1.04maxx,c}],Dashed,Line[{{0,c},{maxx,c}}]}]]

Robin 1984 Theorem 2

{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]+(0.6482/( ((Log[Log[n]])^2)))}/.n->12

N[%]

Solve[((DivisorSigma[1,n])/(n Log[Log[n]]))==Exp[EulerGamma]+(c/( ((Log[Log[n]])^2)))/.n->12]//FullSimplify

N[%,10]

Expand[(1/3) Log[Log[12]] (7-3 (E^EulerGamma) Log[Log[12]])]

data=Subtract@@@Table[{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]+(1/3) Log[Log[12]] (7-3 (E^EulerGamma) Log[Log[12]])(1/( ((Log[Log[n]])^2)))},{n,10^4}];//Timing

Take[data,20]//N

Max[data]//FullSimplify

With[{c=Exp[EulerGamma],maxx=10^4},ListPlot[Subtract@@@Table[{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]+(0.6482/( ((Log[Log[n]])^2)))},{n,2,maxx}]]]

With[{c=Exp[EulerGamma],maxx=10^4},ListPlot[Subtract@@@Table[{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]+(1/3) Log[Log[12]] (7-3 (E^EulerGamma) Log[Log[12]])(1/( ((Log[Log[n]])^2)))},{n,2,maxx}]]]

Robin 1984 Finalement

What are N and x?  N seems to be related to colossally abundant numbers

(EulerGamma+2-Log[4π])//N

2+(EulerGamma+2-Log[4π])-2(Sqrt[2])

%//N

This plot is probably irrelevant since N seems to be related to abundant numbers?

With[{c=Exp[EulerGamma],maxx=10^4},ListPlot[Subtract@@@Table[{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]Exp[((4-2 (Sqrt[2])+EulerGamma-Log[4 π])/(Sqrt[n] Log[n]))]},{n,2,maxx}]]]

Functions website? inequality

Functions website? inequality [wrong and seems to use Robin's abundnant number result for divisor sigma!?]

With[{c=Exp[EulerGamma],maxx=10^4},ListPlot[Subtract@@@Table[{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]+((2(1-(Sqrt[2]))+EulerGamma-Log[4π])/(Sqrt[Log[n]] Log[Log[n]]))},{n,2,maxx}]]]

Valid versions of Functions website (?) inequality as applied to divisor sigma

data=With[{c=Exp[EulerGamma],maxx=10^5},Subtract@@@Table[{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]+((2(Sqrt[2])+EulerGamma-Log[4π])/(Sqrt[Log[n]] Log[Log[n]]))},{n,maxx}]];//Timing

Take[data,12]//N

Max[Drop[data,12]]//FullSimplify

N[%]

ListPlot[Take[data,30]]

With[{c=Exp[EulerGamma],maxx=10^4},ListPlot[Subtract@@@Table[{((DivisorSigma[1,n])/(n Log[Log[n]])),Exp[EulerGamma]+((2(Sqrt[2])+EulerGamma-Log[4π])/(Sqrt[Log[n]] Log[Log[n]]))},{n,2,maxx}]]]

Special values

Squares

Position[DivisorSigma[0,#]&/@Range[200],_?OddQ]//Flatten

Range[14]^2

Position[DivisorSigma[1,#]&/@Range[200],_?OddQ]//Flatten

Select[Union[Flatten[{1,2}#^2&/@Range[200]]],#≤200&]

(DivisorSigma[0, p]):p∈(Primes)

FullSimplify[DivisorSigma[0,p],p∈Primes]

FullSimplify[DivisorSigma[0,p^a],p∈Primes&&a∈Integers&&a>=0]

Prime

(Subscript[σ, 0])(n) is prime when σ(n) is

sigmaprime=Position[Table[DivisorSigma[1,n],{n,10000}],_?PrimeQ]//Flatten

Short[sigma0prime=Flatten[Position[Table[DivisorSigma[0,n],{n,10000}],_?PrimeQ]]]

Complement[sigmaprime,sigma0prime]

Gaussian Integer Divisor Function

Table[DivisorSigma[1,a+b I,GaussianIntegers->True],{a,6},{b,0,6}]//TableForm//TraditionalForm

Table[FactorInteger[1,a+b I,GaussianIntegers->True],{a,6},{b,0,6}]//TableForm//TraditionalForm

FactorInteger[2+2I,GaussianIntegers->True]