View source for Mertens conjecture

{{Short description|Disproved mathematical conjecture}}
[[Image:Congettura Mertens.png|thumb|right|The graph shows the [[Mertens function]] <math>M(n)</math> and the square roots <math>\pm \sqrt{n}</math> for <math>n \le 10,000</math>. After computing these values, Mertens conjectured that the absolute value of <math>M(n)</math> is always bounded by <math>\sqrt{n}</math>. This hypothesis, known as the Mertens conjecture, was disproved in 1985 by [[Andrew Odlyzko]] and [[Herman te Riele]].]]

In [[mathematics]], the '''Mertens conjecture''' is the statement that the [[Mertens function]] <math>M(n)</math> is bounded by <math>\pm\sqrt{n}</math>. Although now disproven, it had been shown to imply the [[Riemann hypothesis]].  It was conjectured by [[Thomas Joannes Stieltjes]], in an 1885 letter to [[Charles Hermite]] (reprinted in {{harvs|txt|last=Stieltjes|authorlink=Thomas Joannes Stieltjes|year=1905}}), and again in print by {{harvs|txt|authorlink=Franz Mertens|last=Mertens|first=Franz|year=1897}}, and disproved by {{harvs|txt|last1=Odlyzko|first1=Andrew|authorlink1=Andrew Odlyzko|last2=te Riele|first2=Herman|authorlink2=Herman te Riele|year=1985}}.
It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.

== Definition ==

In [[number theory]], the [[Mertens function]] is defined as

: <math>M(n) = \sum_{1 \le k \le n} \mu(k),</math>

where &mu;(k) is the [[Möbius function]]; the '''Mertens conjecture''' is that for all ''n'' > 1,

: <math>|M(n)| < \sqrt{n}.</math>

== 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>

== Connection to the Riemann hypothesis ==

The connection to the Riemann hypothesis is based on the [[Dirichlet series]]
for the reciprocal of the [[Riemann zeta function]],

:<math>\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s},</math>

valid in the region <math>\mathcal{Re}(s) > 1</math>. We can rewrite this as a 
[[Stieltjes integral]]

:<math>\frac{1}{\zeta(s)} = \int_0^\infty x^{-s} dM(x)</math>

and after integrating by parts, obtain the reciprocal of the zeta function
as a [[Mellin transform]]

:<math>\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s)
= \int_0^\infty x^{-s} M(x)\, \frac{dx}{x}.</math>

Using the [[Mellin inversion theorem]] we now can express {{mvar|M}} in terms of {{frac|1|{{mvar|&zeta;}}}} as

:<math>M(x) = \frac{1}{2 \pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{x^s}{s \zeta(s)}\,ds</math>

which is valid for {{math|1 < &sigma; < 2}}, and valid for {{math|{{frac|1|2}} < &sigma; < 2}} on the Riemann hypothesis. 
From this, the Mellin transform integral must be convergent, and hence
{{math|''M''(''x'')}} must be {{math|''O''(''x''<sup>e</sup>)}} for every exponent ''e'' greater than {{sfrac|1|2}}. From this it follows that 
:<math>M(x) = O\Big(x^{\tfrac{1}{2} + \epsilon}\Big)</math>

for all positive {{mvar|&epsilon;}} is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that
:<math>M(x) = O\Big(x^\tfrac{1}{2}\Big).</math>

==References==
{{Reflist}}

==Further reading==
* {{cite conference |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]] | book-title=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 }}
* {{cite journal | last1 = Kotnik | first1 = T. | last2 = van de Lune | first2 = J. | year = 2004 | title = On the order of the Mertens function | url = http://www.expmath.org/expmath/volumes/13/13.4/Kotnik.pdf | journal = Experimental Mathematics | volume = 13 | issue = 4 | pages = 473–481 | doi = 10.1080/10586458.2004.10504556 | s2cid = 2093469 | url-status = dead | archive-url = https://web.archive.org/web/20070403143340/http://expmath.org/expmath/volumes/13/13.4/Kotnik.pdf | archive-date = 2007-04-03 }}
* {{cite journal | last1 = Mertens | first1 = F. | year = 1897 | title = Über eine zahlentheoretische Funktion | journal = Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, Abteilung 2a | volume = 106 | pages = 761–830 }}
* {{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 }}
* {{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 }}
* {{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 }}
*{{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
|editor2-first= H.|editor2-last= Bourget|title=Correspondance d'Hermite et Stieltjes|place= Paris|publisher= Gauthier—Villars|year= 1905}}

* {{mathworld|urlname=MertensConjecture|title=Mertens conjecture}}

== External links ==
* {{cite web |title=A Prime Surprise (Mertens Conjecture) |work=[[Numberphile]] |date=January 23, 2020 |via=[[YouTube]] |url=https://www.youtube.com/watch?v=uvMGZb0Suyc  |archive-url=https://ghostarchive.org/varchive/youtube/20211221/uvMGZb0Suyc |archive-date=2021-12-21 |url-status=live}}{{cbignore}}
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[[Category:Analytic number theory]]
[[Category:Disproved conjectures]]