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2004 On the Order of the Mertens Function
Tadej Kotnik, Jan van de Lune
Experiment. Math. 13(4): 473-481 (2004).


We describe a numerical experiment concerning the order of magnitude of {\small $% q(x):=M\left( x\right) /\sqrt{x}$}, where {\small $M(x)$} is the Mertens function (the summatory function of the Möbius function). It is known that, if the Riemann hypothesis is true and all nontrivial zeros of the Riemann zeta-function are simple, {\small $q(x)$} can be approximated by a series of trigonometric functions of {\small $\log x$}. We try to obtain an {\small $\Omega $}-estimate of the order of {\small $q(x)$} by searching for increasingly large extrema of the sum of the first {\small $10^{2}$, $10^{4}$}, and {\small $10^{6}$} terms of this series. Based on the extrema found in the range {\small $10^{4}\leq x\leq 10^{10^{10}}$} we conjecture that {\small $q(x)=\Omega _{\pm }(\sqrt{\log \log \log x})$}.


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Tadej Kotnik. Jan van de Lune. "On the Order of the Mertens Function." Experiment. Math. 13 (4) 473 - 481, 2004.


Published: 2004
First available in Project Euclid: 22 February 2005

zbMATH: 1065.11078
MathSciNet: MR2118272

Primary: 11N56 , 11Y35
Secondary: 11-04

Keywords: Mertens function , Mertens hypothesis , Möbius function

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 4 • 2004
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