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| [[Image:Congettura Mertens.png|thumb|right|The graph shows the [[Mertens function]] ''M''(''n'') and the square roots ±√''n'' for ''n''≤10000. After computing these values Mertens conjectured that the modulus of ''M''(''n'') is always bounded by √''n''. This hypothesis, known as Mertens conjecture, was disproved in 1985 by [[Andrew Odlyzko]] and [[Herman te Riele]]]]
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| In [[mathematics]], the '''Mertens conjecture''' is the incorrect statement that the [[Mertens function]] ''M''(''n'') is bounded by √''n'', which implies the [[Riemann hypothesis]]. It was conjectured by Stieltjes in an 1885 letter to [[Charles Hermite|Hermite]] (reprinted in {{harvnb|Stieltjes|1905}}) and {{harvs|txt|authorlink=Franz Mertens|last=Mertens|year=1897}}, and disproved by {{harvtxt|Odlyzko|te Riele|1985}}.
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| It is a striking example of a mathematical proof contradicting a large amount of computational evidence in favor of a conjecture.
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| == Definition ==
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| In [[number theory]], if we define the [[Mertens function]] as
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| :<math>M(n) = \sum_{1\le k \le n} \mu(k)</math>
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| where μ(k) is the [[Möbius function]], then the '''Mertens conjecture''' is that for all ''n'' > 1,
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| :<math>\left| M(n) \right| < \sqrt { n }.\,</math>
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| == Disproof of the conjecture ==
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| [[Thomas Jan 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 afficionado 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>.)
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| In 1985, [[Andrew Odlyzko]] and [[Herman te Riele]] conditionally proved the Mertens conjecture false: indeed, <math>\liminf m(n) < -1.009</math> and <math>\limsup m(n) > 1.06</math>.<ref>Odlyzko & te Riele (1985)</ref><ref name=HBI1889>Sandor et al (2006) pp.188–189</ref> It was later shown that the first [[counterexample]] appears below exp(3.21{{e|64}}) ([[János Pintz|Pintz]] 1987) but above 10<sup>14</sup> (Kotnik and Van de Lune 2004). The upper bound has since been lowered to exp(1.59{{e|40}}) (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|lc=on}}). The [[law of the iterated logarithm]] states that if
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| 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'')<sup>1/2</sup>, which suggests that the order of growth of ''m''(''n'') might be somewhere around (log log ''n'')<sup>1/2</sup>. 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 <math>(\log{\log{\log{n}}})^{5/4}</math>, 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 of the Riemann zeta function.<ref>{{cite web|url=http://www.cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf|title=The distribution of the summatory function of the Möbius function}}</ref>
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| In 1979 Cohen and Dress found the largest known value of <math>m(n) \approx 0.570591</math> for M(7766842813) = 50286. In 2003 Kotnik and van de Lune extended the search to ''n'' = 10<sup>14</sup> 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''.<ref>Kotnik & te Riele (2006)</ref>
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| == Connection to the Riemann hypothesis ==
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| The connection to the Riemann hypothesis is based on the [[Dirichlet series]]
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| for the reciprocal of the [[Riemann zeta function]],
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| :<math>\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s},</math> | |
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| valid in the region <math>\Re(s) > 1</math>. We can rewrite this as a
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| [[Stieltjes integral]]
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| :<math>\frac{1}{\zeta(s)} = \int_0^{\infty} x^{-s}\,dM(x)</math>
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| and after integrating by parts, obtain the reciprocal of the zeta function
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| as a [[Mellin transform]]
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| :<math>\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s)
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| = \int_0^\infty x^{-s} M(x)\, \frac{dx}{x}.</math>
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| Using the [[Mellin inversion theorem]] we now can express ''M'' in terms of
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| 1/ζ as
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| :<math>M(x) = \frac{1}{2 \pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{x^s}{s \zeta(s)}\, ds</math> | |
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| which is valid for 1 < σ < 2, and valid for 1/2 < σ < 2 on the Riemann hypothesis.
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| From this, the Mellin transform integral must be convergent, and hence
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| ''M''(''x'') must be ''O''(''x''<sup>e</sup>) for every exponent ''e'' greater than
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| 1/2. From this it follows that
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| :<math>M(x) = O(x^{\frac12+\epsilon})</math>
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| for all positive ε is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that
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| :<math>M(x) = O(x^\frac12)</math>.
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| ==References==
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| <references />
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| * {{cite conference |url= http://www.springerlink.com/content/q0717243567v503t/ |title=The Mertens Conjecture Revisited | first1=Tadej | last1=Kotnik | first2=Herman | last2=te Riele | author2-link=Herman te Riele | year= 2006 | series=Lecture Notes in Computer Science | volume=4076 | publisher=[[Springer-Verlag]] | booktitle=Algorithmic number theory. 7th international symposium, ANTS-VII, Berlin, Germany, July 23--28, 2006. Proceedings | editor1-last=Hess | editor1-first=Florian | pages=156–167 | location=Berlin | doi=10.1007/11792086_12 | zbl=1143.11345 | isbn=3-540-36075-1 }}
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| * T. Kotnik and J. van de Lune (2004), "[http://www.expmath.org/expmath/volumes/13/13.4/Kotnik.pdf On the order of the Mertens function]", ''Experimental Mathematics'' '''13''', pp. 473–481
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| * F. Mertens (1897), "Über eine zahlentheoretische Funktion", ''Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, Abteilung 2a'', '''106''', pp. 761–830.
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| * {{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 | id={{MathSciNet | id = 783538}} | year=1985 | journal=[[Journal für die reine und angewandte Mathematik]] | volume=357 | pages=138–160 | zbl=0544.10047 | issn=0075-4102 }}
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| * J. Pintz (1987), "An effective disproof of the Mertens conjecture", ''Astérisque'' '''147-148''', pp. 325–333.
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| * {{citation | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | pages=187–189 }}
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| *{{citation|first= T. J.|last= Stieltjes|chapter= Lettre a Hermite de 11 juillet 1885, Lettre #79|pages= 160–164 |editor-first=B.|editor-last= Baillaud
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| |editor2-first= H.|editor2-last= Bourget|title=Correspondance d'Hermite et Stieltjes|place= Paris|publisher= Gauthier—Villars|year= 1905}}
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| * {{mathworld|urlname=MertensConjecture|title=Mertens conjecture}}
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| [[Category:Number theory]]
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| [[Category:Disproved conjectures]]
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