Mertens Conjecture

Author

Eric W. Weisstein
July 1, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

Mertens[x_]:=Plus@@MoebiusMu[Range[1,x]]
Mertens[{1,x_}]:=Rest@FoldList[Plus,0,MoebiusMu[Range[1,x]]]

Plot

<<Graphics`Colors`
<<Utilities`Typesetting`

With[{n=10^5},
ListPlot[Mertens[{1,n}]/Sqrt[Range[n]],PlotStyle->Red,PlotJoined->True,AxesLabel->TraditionalForm/@{x,InlineFraction[M[x],Sqrt[x]]}]
]//Timing

Show[
Block[{$DisplayFunction=Identity},
{ListPlot[Mertens[{1,10000}],AxesLabel->TraditionalForm/@{n,M[n]},
PlotJoined->True,PlotStyle->Red],
Plot[{-Sqrt[x],Sqrt[x]},{x,0,10000},PlotStyle->{{Blue}}]
}
]
]