The Riemann hypothesis and pseudorandom features of the Möbius sequence

Abstract:

A study of the cumulative sums of the Möbius function on the Atlas Computer of the Science Research Council has revealed certain statistical properties which lead the authors to make a number of conjectures. One of these is that any conjecture of the Mertens type, viz. \[ |M(N)| = \left | {\sum \limits _{n = 1}^N {\mu (n)} } \right | < k\left ( {\surd N} \right )\] where $k$ is any positive constant, is false, and indeed the authors conjecture that \[ {\text {Lim}}\sup \left \{ {M(x){{(x\log \log x)}^{ - 1/2}}} \right \} = {{\surd \left ( {12} \right )} \left / {\vphantom {{\surd \left ( {12} \right )} \pi }} \right . \pi }\] .