Eine empirische Untersuchung zur Mertensschen Funktion (Q1131279)

Elaborate calculations performed by the author on the computer of the University of Heidelberg give the behaviour of the functions \(M_1(x)/\sqrt x\), \(M_2(x)/x\) for many values of \(x\). Here \(M_1(x) = \sum_{n\le x} \mu(n)\), \(M_2(x) = \sum_{n\le x} M_1(n)\), and \(\mu(n)\) is the Möbius function. Specific values of \(x\) are given for which \(\vert M_1(x)/\sqrt x \vert > \tfrac12\) (e.g. \(x = 7.76\cdot 10^9)\) and the computation indicates that the conjectured relationship \(M_2(x) = O(x \log x)\) is probably untrue.