Editing Mertens conjecture (section)

== Disproof of the conjecture ==
[[Thomas Joannes Stieltjes|Stieltjes]] claimed in 1885 to have proven a weaker result, namely that <math>m(n) := M(n)/\sqrt{n}</math> was [[Bounded function|bounded]], but did not publish a proof.<ref>{{cite book |editor1-last=Borwein |editor1-first=Peter |editor1-link=Peter Borwein |editor2-last=Choi |editor2-first=Stephen |editor3-last=Rooney |editor3-first=Brendan |editor4-last=Weirathmueller |editor4-first=Andrea |title=The Riemann hypothesis. A resource for the aficionado and virtuoso alike |series=CMS Books in Mathematics |location=New York, NY | publisher=[[Springer-Verlag]] |year=2007 |isbn=978-0-387-72125-5 |zbl=1132.11047 |page=69}}</ref> (In terms of <math>m(n)</math>, the Mertens conjecture is that <math> -1 < m(n) < 1 </math>.)

In 1985, [[Andrew Odlyzko]] and [[Herman te Riele]] proved the Mertens conjecture false using the [[Lenstra–Lenstra–Lovász lattice basis reduction algorithm]]:<ref name= disproof>{{Citation | last1=Odlyzko | first1=A. M. | author1-link=Andrew Odlyzko | last2=te Riele | first2=H. J. J. | author2-link=Herman te Riele | title=Disproof of the Mertens conjecture | url=http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf | doi=10.1515/crll.1985.357.138 |mr=783538 | year=1985 | journal=[[Journal für die reine und angewandte Mathematik]] | volume=1985 | issue=357 | pages=138–160 | zbl=0544.10047 | s2cid=13016831 | issn=0075-4102 }}</ref><ref name=HBI1889>Sandor et al (2006) pp. 188–189.</ref>
: <math>\liminf m(n) < -1.009 </math> {{pad|4}} and {{pad|4}} <math> \limsup m(n) > 1.06.</math>
It was later shown that the first [[counterexample]] appears below <math>e^{3.21\times10^{64}} \approx 10^{1.39\times10^{64}}</math><ref>{{cite journal | last1 = Pintz | first1 = J. | year = 1987 | title = An effective disproof of the Mertens conjecture | url = http://www.numdam.org/article/AST_1987__147-148__325_0.pdf | journal = [[Astérisque]] | volume = 147–148 | pages = 325–333 | zbl=0623.10031 }}
</ref> but above 10<sup>16</sup>.<ref name=H16>{{cite arXiv |last=Hurst |first=Greg |date=2016 |title=Computations of the Mertens function and improved bounds on the Mertens conjecture |eprint=1610.08551 |class=math.NT}}</ref> The upper bound has since been lowered to <math>e^{1.59\times10^{40}}</math><ref>Kotnik and Te Riele (2006).</ref> or approximately <math>10^{6.91\times10^{39}},</math> and then again to <math>e^{1.017\times10^{29}} \approx 10^{4.416\times10^{28}}</math>.<ref>{{Cite arXiv |last1=Rozmarynowycz |first1=John |last2=Kim |first2=Seungki |year=2023 |title=A New Upper Bound On the Smallest Counterexample To The Mertens Conjecture |class=math.NT |eprint=2305.00345 }}</ref> In 2024, Seungki Kim and [[:fr:Phong Nguyen|Phong Nguyen]] lowered the bound to <math>e^{1.96\times10^{19}} \approx 10^{8.512\times10^{18}}</math>,<ref>{{Cite web |last1=Seungki |first1=Kim |last2=Phong |first2=Nguyen |year=2024 |title=On counterexamples to the Mertens conjecture |url=https://antsmath.org/ANTSXVI/papers/KimNguyen.pdf}}</ref> but no ''explicit'' counterexample is known.

The [[law of the iterated logarithm]] states that if {{mvar|&mu;}} is replaced by a random sequence of +1s and &minus;1s then the order of growth of the partial sum of the first {{mvar|n}} terms is (with probability 1) about {{nowrap|{{math|{{sqrt| ''n'' log log ''n''}}}},}} which suggests that the order of growth of {{math|''m''(''n'')}} might be somewhere around {{math|{{sqrt|log log ''n''}}}}. The actual order of growth may be somewhat smaller;  in the early 1990s Steve Gonek conjectured<ref name=NGonMertens /> that the order of growth of {{math|''m''(''n'')}} was <math>(\log\log\log n)^{5/4},</math> which was affirmed by Ng (2004), based on a heuristic argument, that assumed the Riemann hypothesis and certain conjectures about the averaged behavior of zeros of the Riemann zeta function.<ref name=NGonMertens>{{cite web |url=http://www.cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf |title=The distribution of the summatory function of the Möbius function |first=Nathan |last=Ng |year=2004}}</ref>

In 1979, Cohen and Dress<ref>Cohen, H. and Dress, F. 1979. “Calcul numérique de Mx)” 11–13. [Cohen et Dress 1979], Rapport, de I'ATP A12311 ≪ Informatique 1975 ≫</ref> found the largest known value of <math>m(n) \approx 0.570591</math> for {{math|''M''(7766842813) {{=}} 50286,}} and in 2011, Kuznetsov found the largest known negative value (largest in the sense of [[absolute value]]) <math>m(n) \approx -0.585768</math> for {{math|''M''(11609864264058592345) {{=}} &minus;1995900927.}}<ref>{{cite arXiv |last=Kuznetsov |first=Eugene |date=2011 |title=Computing the Mertens function on a GPU |eprint=1108.0135 |class=math.NT}}</ref> In 2016, Hurst computed {{math|''M''(''n'')}} for every {{math|''n'' ≤ 10<sup>16</sup>}} but did not find larger values of {{math|''m''(''n'')}}.<ref name=H16></ref>

In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of {{mvar|n}} for which {{nowrap|{{math|''m''(''n'') > 1.2184}},}} but without giving any specific value for such an {{mvar|n}}.<ref>Kotnik & te Riele (2006).</ref> In 2016, Hurst made further improvements by showing
: <math>\liminf m(n) < -1.837625 </math> {{pad|4}} and {{pad|4}} <math> \limsup m(n) > 1.826054.</math>