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    Mertens Conjecture

    Statement

    The Mertens conjecture, now refuted, posited that for all integers n>1, left bracketing bar M(n) right bracketing bar <sqrt(n), where M(n) is the summatory Möbius function.

    Solution

    False

    Formal statement

    for all _(n, n element Z ∧ n>1) left bracketing bar sum_(k=1)^nμ(k) right bracketing bar <sqrt(n)

    History

    formulation date | 1885 (139 years ago) formulator | Thomas Jan Stieltjes status | refuted proof date | 1985 (100 years later) (39 years ago) provers | Andrew Odlyzko | Herman J.J. te Riele

    Associated equation

    left bracketing bar sum_(k=1)^nμ(k) right bracketing bar <sqrt(n)

    Current evidence

    The value of the smallest counterexample is still unknown but known to exceed 1×10^14.

    Classes

    refuted conjectures | solved mathematics problems

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