The Mertens Conjecture Revisited

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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 \(M(x)/\sqrt{x}=\Omega_{\pm}(\sqrt{\log\log\log x})\).