The Mertens conjecture revisited

Abstract

Let M(x) = Σ1≤n≤xμ(n) where μ,(n) is the Möbius function. The Mertens conjecture that |M(x)|/ √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 lim sup M(x)/√x is raised to 1.218, and the known upper bound -1.009 for lim inf M(x)/√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 × 10 64) to exp(1.59 × 1040). Finally, new numerical evidence is presented for the conjecture that M(x)/ √x = Ω ±(√log log log x). © Springer-Verlag Berlin Heidelberg 2006.