Disproof of the Mertens conjecture.

@article{Odlyzko1984DisproofOT,
  title={Disproof of the Mertens conjecture.},
  author={A. Odlyzko and H. J. Riele},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={1984},
  volume={1985},
  pages={138 - 160}
}
  • A. Odlyzko, H. J. Riele
  • Published 1984
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
The Mertens conjecture states that  M(x)  1, where M(x) = n ≤ x Σ μ(n) , and μ(n) is the Mo bius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesis. This paper disproves the Mertens conjecture by showing that x → ∞ lim sup M(x) x − ⁄2 > 1. 06 . The disproof relies on extensive computations with the zeros of the zeta function, and does not provide an explicit… Expand

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References

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