Stieltjes Constants

Author

Eric W. Weisstein
January 16, 2015

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/StieltjesConstants.nb.

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

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

Plot

<<Utilities`Plot`
<<Utilities`Typesetting`

IntegerListPlot[StieltjesGamma[Range[30]],PlotRange->{-.005,.01},AxesLabel->TraditionalForm/@{n,StieltjesGamma[n]}]

(gammas=N[StieltjesGamma[Range[0,800]],5]);//Timing

Show[GraphicsArray[Block[{$DisplayFunction=Identity},
{
ListPlot[MapIndexed[{#2[[1]]-1,Log[Abs[#]]}&,Take[gammas,200]],AxesLabel->TraditionalForm/@{n,Log[Abs[StieltjesGamma[n]]]},PlotStyle->{Red,PointSize[.015]}],
ListPlot[MapIndexed[{600+#2[[1]],Log[Abs[#]]}&,Take[gammas,{601,650}]],AxesLabel->TraditionalForm/@{n,Log[Abs[StieltjesGamma[n]]]/n},PlotStyle->{Red,PointSize[.015]}]
}]],ImageSize->500]

Signs

Take[Sign[gammas],10]

Length/@Split[Sign[gammas]]

ListPlot[Length/@Split[Sign[gammas]],PlotStyle->Red]

Limits

(StieltjesGamma[n])==(lim_(z->1)) ((((-1))^n) (((∂^n)(Zeta[z]))/(∂(z^n)))-((n!)/(((z-1))^(n+1))))

Table[Limit[(-1)^n*D[Zeta[z],{z,n}]-n!/(z-1)^(n+1),z->1],{n,0,10}]

Inequalities

(StieltjesGamma[n])<(((n-1)!)/(π^n)) ((Piecewise[{{4, (Mod[n, 2])==0}, {2, (Mod[n, 2])==1}}]))

ListPlot[Table[StieltjesGamma[n]-(((n-1)!)/(π^n))Piecewise[{{4,Mod[n,2]==0},{2,Mod[n,2]==1}}],{n,10}],Joined->True,PlotStyle->Red,PlotRange->All,AxesOrigin->{0,0},AxesLabel->{n,(StieltjesGamma[n])<(((n-1)!)/(π^n)) ((Piecewise[{{4, (Mod[n, 2])==0}, {2, (Mod[n, 2])==1}}]))}]

Position[Table[StieltjesGamma[n]<(((n-1)!)/(π^n))Piecewise[{{4,Mod[n,2]==0},{2,Mod[n,2]==1}}],{n,20}],False]

(Abs[(StieltjesGamma[n])])<((exp(n log(log(n))))/(10^4))

Position[Table[Abs[StieltjesGamma[n]]<(10^(-4 ))Exp[n Log[Log[n]]],{n,50}],False]//Flatten

ListPlot[Table[Abs[StieltjesGamma[n]]-(10^(-4 ))Exp[n Log[Log[n]]],{n,10}],PlotRange->All,Joined->True,PlotStyle->{Thick,Red},AxesLabel->Abs[StieltjesGamma[n]]-(10^(-4. ))Exp[n Log[Log[n]]]]

Sums

(∑_(n=0)^∞)-((((((-1))^n) (n+1) (StieltjesGamma[n+1])) (((z-1))^n))/((n+1)!))==ζ'(z)+(1/(((z-1))^2))

Series[Zeta'[z],{z,1,2}]

SeriesCoefficient[Zeta'[z],{z,1,n}]

-(((((-1))^n) (1+n) StieltjesGamma[1+n])/((1+n)!))/.n->0

Sum[-(((((-1))^n) (1+n) StieltjesGamma[1+n])/((1+n)!))(((z-1))^n),{n,0,5}]

-(1/(((z-1))^2))+Sum[-(((((-1))^n) (1+n) StieltjesGamma[1+n])/((1+n)!))(((z-1))^n),{n,0,∞}]//Timing

-(1/(((z-1))^2))+Sum[-(((((-1))^n) (1+n) StieltjesGamma[1+n])/((1+n)!))(((z-1))^n),{n,∞}]//Timing

SumConvergence[(((((-1))^n) (1+n) (((-1+z))^n) StieltjesGamma[1+n])/((1+n)!)),n]

With[{z=1.1},{Sum[-(((((-1))^n) (1+n) StieltjesGamma[1+n])/((1+n)!))(((z-1))^n),{n,0,100}],(1/(((z-1))^2))+Zeta'[z]}]

Table[Normal[Series[D[Zeta[z],{z,n}],{z,1,2}]]-(((-1))^n)n!/(((z-1))^(n+1)),{n,3}]/.z->1

(∑_k^∞)(((((-1))^k) (((s-1))^(k-1)) (StieltjesGamma[k]))/(k!))==(((Zeta[s])-(1/(s-1))-ℽ)/(s-1))

Sum[1/k! StieltjesGamma[k] (-1)^k (s-1)^(k-1),{k,Infinity}]

ζ(z)=(∑_(n=0)^∞)((((-1))^n)/(n!)) (Subscript[γ, n]) (((z-1))^n)+(1/(z-1))

V6

1/(z-1)+Sum[(-1)^n/n! StieltjesGamma[n](z-1)^n,{n,0,Infinity}]

V7

1/(z-1)+Sum[(-1)^n/n! StieltjesGamma[n](z-1)^n,{n,0,Infinity}]

ζ(z+1)-(1/z)=(∑_(k=0)^∞)(((((-1))^k) (Subscript[γ, k]) (z^k))/(k!))

V6

Sum[(-1)^k/k!*StieltjesGamma[k]z^k, {k, 0, Infinity}]

V7

Sum[(-1)^k/k!*StieltjesGamma[k]z^k, {k, 0, Infinity}]

(Subscript[γ, n])==(((ln^n)(2))/(n+1)) (∑_k^∞)(((((-1))^k) (Subscript[B, n+1])((Subscript[log, 2])(k)))/k)

Log[2]^n/(n+1)Sum[(-1)^k/k BernoulliB[n+1,Log[2,k]],{k,∞}]//Timing

Table[(((Log[2])^n)/(n+1))Sum[(((((-1))^k)BernoulliB[n+1,Log[2,k]])/k),{k,∞}],{n,10}]

(((Log[2])^n)/(n+1))Sum[(((((-1))^k)BernoulliB[n+1,Log[2,k]])/k),{k,∞}]//Timing

N[Table[{StieltjesGamma[n],(((Log[2])^n)/(n+1))Sum[(((((-1))^k)BernoulliB[n+1,Log[2,k]])/k),{k,(10^5)}]},{n,0,3}]]//Timing

(StieltjesGamma[n])==n! (log^n)(2) (∑_m^(n+1))((((-1))^(m-1))/(m!)) (∑_k^∞)(((((-1))^k) ((Floor[(Subscript[log, 2])(k)])^m) (BernoulliB2[-m+n+1, (Subscript[log, 2])(k)]))/k)

N[Table[{StieltjesGamma[n],n!*Log[2]^n*Sum[((-1)^(m-1)/m!)*Sum[((-1)^k/k)*Floor[Log[2,k]]^m*BernoulliB[1+n-m,Log[2,k]],{k,10^5}],{m,n+1}]},{n,0,3}]]//Timing

Table[{StieltjesGamma[n]//N,n!*Log[2]^n*Sum[((-1)^(m-1)/m!)((-1)^k/k)*Floor[Log[2,k]]^m*BernoulliB[1+n-m,Log[2,k]],{k,10^5},{m,n+1}]//N[#,10]&},{n,0,2}]//Timing

Table[{StieltjesGamma[n]//N,n!*Log[2]^n*Sum[((-1)^(m-1)/m!)((-1)^k/k)*Floor[Log[2,k]]^m*BernoulliB[1+n-m,Log[2,k]],{k,10^6},{m,n+1}]//N[#,10]&},{n,0,2}]//Timing

Table[{StieltjesGamma[n]//N,n!*Log[2]^n*NSum[((-1)^(m-1)/m!)((-1)^k/k)*Floor[Log[2,k]]^m*BernoulliB[Round[1+n-m],Log[2,k]],{k,∞},{m,n+1}]},{n,0,2}]//Timing

(∑_(k=0)^∞)((StieltjesGamma[k+n])/(k!))==(((-1))^n) (n!+Derivative[n][ζ](0))/;n∈ℤ∧n≥0

Sum[StieltjesGamma[k+n]/k!,{k,0,Infinity}]==(-1)^n*(n!+Derivative[n][Zeta][0])/;Element[n,Integers]&&n≥0

Sum[StieltjesGamma[k+n]/k!,{k,0,Infinity}]

Table[{Sum[StieltjesGamma[k+n]/k!,{k,0,100}],(-1)^n*(n!+Derivative[n][Zeta][0])},{n,5}]//N

(Subscript[γ, 0])

StieltjesGamma[0]

StieltjesGamma[0]//N

n! (log^n)(2) (∑_m^(n+1))((((-1))^(m-1))/(m!)) (∑_k^∞)(((((-1))^k) ((Floor[(Subscript[log, 2])(k)])^m) (BernoulliB2[-m+n+1, (Subscript[log, 2])(k)]))/k)

With[{n=0},n! Log[2]^n Sum[(-1)^(m-1)/m! Sum[(-1)^k/k Floor[Log[2,k]]^m BernoulliB[1+n-m,Log[2,k]],{k,∞}],{m,n+1}]]

(((log^n)(2))/(n+1)) (∑_k^∞)(((((-1))^k) (BernoulliB2[n+1, (Subscript[log, 2])(k)]))/k)

With[{n=0},Log[2]^n/(n+1)Sum[(-1)^k/k BernoulliB[n+1,Log[2,k]],{k,∞}]]

(Subscript[γ, 1])

StieltjesGamma[1]//N

n! (log^n)(2) (∑_m^(n+1))((((-1))^(m-1))/(m!)) (∑_k^∞)(((((-1))^k) ((Floor[(Subscript[log, 2])(k)])^m) (BernoulliB2[-m+n+1, (Subscript[log, 2])(k)]))/k)

With[{n=1},n! Log[2]^n Sum[(-1)^(m-1)/m! Sum[(-1)^k/k Floor[Log[2,k]]^m BernoulliB[1+n-m,Log[2,k]],{k,∞}],{m,n+1}]]

(((log^n)(2))/(n+1)) (∑_k^∞)(((((-1))^k) (BernoulliB2[n+1, (Subscript[log, 2])(k)]))/k)

With[{n=1},Log[2]^n/(n+1)Sum[(-1)^k/k BernoulliB[n+1,Log[2,k]],{k,∞}]]

(StieltjesGamma[1])==(1/(2 log(2)))(∑_(k=1)^∞)(((((-1))^k) (log^2)(k))/k)+(((log^2)(2))/6)-(1/2) ℽ log(2)

Sum[(((((-1))^k)((Log[k])^2))/k),{k,1,∞}]

(1/(2Log[2]))Sum[(((((-1))^k)((Log[k])^2))/k),{k,1,∞}]-(1/2)EulerGamma Log[2]+(1/6)((Log[2])^2)//FullSimplify

(∑_(n=1)^x)(1/n)ln((x/n))==(HarmonicNumber[x]) (lnx)-(StieltjesGamma[1])+(StieltjesGamma2[1, x+1])

V4.2

Sum[Log[x/n]/n,{n,x}]//InputForm

...wrong...

V10

Sum[Log[x/n]/n,{n,x}]

TraditionalForm[%]

FullSimplify[Series[%%,{x,∞,1}],x>0]

With[{x=10},{Sum[Log[x/n]/n,{n,x}],HarmonicNumber[x] Log[x]-StieltjesGamma[1]+StieltjesGamma[1,1+x]}]//N

(lim_(y->∞)) y (y+Im((Zeta[1+(i/y)])))

V5

Limit[y(Im[Zeta[1+I/y]]+y),y->∞]

N[y(Im[Zeta[1+I/y]]+y)/.y->99999999,20]

V7

Limit[y(Im[Zeta[1+I/y]]+y),y->∞]

Ellison-Mendès-Frace form of (Subscript[γ, 1])

f[x_]:=FullSimplify[Sum[Log[x/n]/n,{n,x}]-Log[x]^2/2-EulerGamma Log[x]]

f[x]

f[x] /. x->10.

f[10.]

g[x_,opts___]:=
NSum[Log[x/n]/n,{n,x},AccuracyGoal->Infinity,WorkingPrecision->30,MaxIterations->10]-Log[x]^2/2-EulerGamma Log[x]

({#,g[#]}&/@Table[10^n,{n,7}])//ColumnForm

N[StieltjesGamma[1],20]

Numerical Values

Table[{n,N[StieltjesGamma[n],20]},{n,10}]//TableForm