Mertens Conjecture - Disproof of The Conjecture

Disproof of The Conjecture

Stieltjes claimed in 1885 to have proven a weaker result, namely that was bounded, but did not publish a proof.

In 1985, Andrew Odlyzko and Herman te Riele proved the Mertens conjecture false. It was later shown that the first counterexample appears below exp(3.21×1064) (Pintz 1987) but above 1014 (Kotnik and Van de Lune 2004). The upper bound has since been lowered to exp(1.59×1040) (Kotnik and Te Riele 2006), but no counterexample is explicitly known. The boundedness claim made by Stieltjes, while remarked upon as "very unlikely" in the 1985 paper cited above, has not been disproven (as of 2009). The law of the iterated logarithm states that if μ is replaced by a random sequence of 1s and −1s then the order of growth of the partial sum of the first n terms is (with probability 1) about (n log log n)1/2, which suggests that the order of growth of m(n) might be somewhere around (log log n)1/2. The actual order of growth may be somewhat smaller; it was conjectured by Gonek in the early 1990s that the order of growth of m(n) was, which was also conjectured by Ng (2004), based on a heuristic argument assuming the Riemann hypothesis and certain conjectures about the averaged behavior of zeros the Riemann zeta function.

In 1979 Cohen and Dress found the largest known value of m(n) =~ 0.570591 for M(7766842813) = 50286. In 2003 Kotnik and van de Lune extended the search to n = 1014 but did not find larger values. In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of n for which m(n)>1.2184, but without giving any specific such value for n.