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: where μ(k) is the Möbius Function . Because the Möbius function has only the return values -1, 0 and +1, it's obvious that the Mertens function moves slowly and that there is no ''x'' such that ''M''(''x'') > ''x''. The Mertens Conjecture goes even further, stating that there is no ''x'' where the absolute value of the Mertens function exceeds the square root of ''x''. The Mertens conjecture was disproven in 1985 . However, the Riemann Hypothesis is equivalent to a weaker conjecture on the growth of ''M''(''x''), namely . Since high values for M grow at least as fast as the square root of x, this puts a rather tight bound on its rate of growth. EXTERNAL LINKS
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