The Mertens Conjecture Revisited

Abstract

Let M(x)=∑_{1 ≤ n ≤ x} μ(n) where μ(n) is the Möbius function. The Mertens conjecture that $|M(x)|/\sqrt{x}<1$ for all x>1 was disproved in 1985 by Odlyzko and te Riele [13]. In the present paper, the known lower bound 1.06 for $\limsup M(x)/\sqrt{x}$ is raised to 1.218, and the known upper bound –1.009 for $\liminf M(x)/\sqrt{x}$ is lowered to –1.229. In addition, the explicit upper bound of Pintz [14] on the smallest number for which the Mertens conjecture is false, is reduced from $\exp(3.21\times10^{64})$ to $\exp(1.59\times10^{40})$. Finally, new numerical evidence is presented for the conjecture that $M(x)/\sqrt{x}=\Omega_{\pm}(\sqrt{\log\log\log x})$.