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The Mertens Conjecture Revisited

  • Tadej Kotnik
  • Herman te Riele
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)

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

Keywords

Numerical Evidence Riemann Zeta Function Riemann Hypothesis Decimal Digit Lattice Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Tadej Kotnik
    • 1
  • Herman te Riele
    • 2
  1. 1.Faculty of Electrical EngineeringUniversity of LjubljanaSlovenia
  2. 2.CWIAmsterdamThe Netherlands

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