The distribution of the summatory function of the Möbius function

Abstract

The summatory function of the Möbius function is denoted $M(x)$. In this article we deduce 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. Assuming these conjectures, we show that $M(x)$, when appropriately normalized, possesses a limiting distribution, and also that a strong form of the weak Mertens conjecture is true. Finally, we speculate on the lower order of $M(x)$ by studying the constructed distribution function.