1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking

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/2 M(e y) 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and large negative as a constant times ± √ x(log log log x) 5 4 infinitely often, thus providing evidence for an unpublished conjecture of Gonek’s. 1

Discrete moments of the Riemann zeta function were studied by Gonek and Hejhal in the 1980’s. They independently formulated a conjecture concerning the size of these moments. In 1999, Hughes, Keating, and O’Connell, by employing a random matrix model, made this conjecture more precise. Subject to the Riemann hypothesis, we establish upper and lower bounds of the correct order of magnitude in the case of the fourth moment. 1