Experimental Mathematics

On the Order of the Mertens Function

Tadej Kotnik and Jan van de Lune

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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 $q(x)=\Omega _{\pm }(\sqrt{\log \log \log x})$}.

Article information

Experiment. Math., Volume 13, Issue 4 (2004), 473-481.

First available in Project Euclid: 22 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N56: Rate of growth of arithmetic functions 11Y35: Analytic computations
Secondary: 11-04: Explicit machine computation and programs (not the theory of computation or programming)

Mertens function Möbius function Mertens hypothesis


Kotnik, Tadej; van de Lune, Jan. On the Order of the Mertens Function. Experiment. Math. 13 (2004), no. 4, 473--481. https://projecteuclid.org/euclid.em/1109106439

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