On the Order of the Mertens Function

Abstract

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