The distribution of the summatory function of the Möbius function

Let the summatory function of the Möbius function be denoted M(x). We deduce in this article conditional results concerning M(x) assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. The main results shown are that the weak Mertens conjecture and the existence of a limiting distribution of e−y/2M(ey) are consequences of the aforementioned conjectures. By probabilistic techniques, we present an argument that suggests M(x) grows as large positive and large negative as a constant times ±√x(log log log x) 54 infinitely often, thus providing evidence for an unpublished conjecture of Gonek’s.