Riemann Hypothesis

Author

Eric W. Weisstein
January 4, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/RiemannHypothesis.nb.

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

"Short proof" by
Michael Trott
June 9, 2004

Lagarias

Inequality

sigma=Table[DivisorSigma[1,n],{n,1000}];//Timing

harm=Table[#+Exp[#]Log[#]&[HarmonicNumber[n]],{n,1000}];//Timing

Inequality check

And@@(LessEqual/@Transpose[{sigma,harm}])

Small values

small=Position[harm-sigma,#]&/@Take[Sort[harm-sigma,Less],100]

Flatten[small]

N[harm-sigma][[Flatten[small]]]

Plot

<<Graphics`MultipleListPlot`

Show[GraphicsArray[{
MultipleListPlot[sigma,harm,PlotJoined->True,SymbolShape->None,PlotStyle->{{},RGBColor[1,0,0]},
AxesLabel->TraditionalForm/@{n,DivisorSigma[1,n]},
DisplayFunction->Identity],
ListPlot[harm-sigma,AxesLabel->TraditionalForm/@{n,#+Exp[#]ln[#]&[HarmonicNumber[n]]-DivisorSigma[1,n]},
DisplayFunction->Identity]
}]]

Limits

GramPoint[200000001,WorkingPrecision->20]

GramPoint[10^12,WorkingPrecision->10]

GramPoint[10^13,WorkingPrecision->10]

Short Proof

N.B. **this is a joke***

Exists[{s,t},
(* a zero in the strip,not on the critical line *)
Element[{s,t},Reals]&&s>0&&s≠1/2,Zeta[s+I t]==0]//TrueQ