Gronwall's Theorem

Author

Eric W. Weisstein
August 29, 2019

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

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

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

Plot

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}}]}]]

Supremum

l=Table[((DivisorSigma[1,n])/(n Log[Log[N[n]]])),{n,10^6}];//Timing

Through[{Min,Mean,Max}[Take[l,-10000]]]

Exp[EulerGamma]//N

ListPlot[Max/@Partition[l,1000],Joined->True]