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The Mertens conjecture revisited. (English) Zbl 1143.11345
Hess, Florian (ed.) et al., Algorithmic number theory. 7th international symposium, ANTS-VII, Berlin, Germany, July 23–28, 2006. Proceedings. Berlin: Springer (ISBN 3-540-36075-1/pbk). Lecture Notes in Computer Science 4076, 156-167 (2006).
Summary: Let M(x)= 1nx μ(n) where μ(n) is the Möbius function. The Mertens conjecture that |M(x)|/x<1 for all x>1 was disproved by A. M. Odlyzko and H. J. J. te Riele [J. Reine Angew. Math. 357, 138–160 (1985; Zbl 0544.10047)]. In the present paper, the known lower bound 1.06 for lim supM(x)/x is raised to 1.218, and the known upper bound -1·009 for lim infM(x)/x is lowered to -1·229. In addition, the explicit upper bound of J. Pintz [An effective disproof of the Mertens conjecture. Astérisque, 147–148, 325–333 (1987; Zbl 0623.10031)] on the smallest number for which the Mertens conjecture is false, is reduced from exp(3·21×10 64 ) to exp(1·59×10 40 ). Finally, new numerical evidence is presented for the conjecture that M(x)/x=Ω ± (logloglogx).
MSC:
11N37Asymptotic results on arithmetic functions
11Y35Analytic computations