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In mathematics, the Mertens conjecture is the statement that the Mertens function is bounded by . 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 Stieltjes), and again in print by Franz Mertens, and disproved by Andrew Odlyzko and Herman te Riele.It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.

Attributes	Values
rdf:type	Thing yago:WikicatConjectures yago:Abstraction100002137 yago:Cognition100023271 yago:Concept105835747 yago:Content105809192 yago:Hypothesis105888929 yago:Idea105833840 yago:PsychologicalFeature100023100 yago:Speculation105891783 yago:WikicatDisprovedConjectures
rdfs:label	حدسية ميرتنز (ar) Mertenssche Vermutung (de) Conjetura de Mertens (es) Conjecture de Mertens (fr) Mertens conjecture (en) Vermoeden van Mertens (nl) 梅滕斯猜想 (zh) Mertens förmodan (sv)
rdfs:comment	حدسية ميرتنز في نظرية الأعداد تنص على أن القيمة المطلقة لدالة ميرتنز لا تتجاوز الجذر المربع ل n. وهي حدسية خاطئة. وضعت عام 1897 وثبت خطأها في عام 1985 وسميت هاته الحدسية هكذا نسبة لعالم الرياضيات فرانز ميرتنز. (ar) En matemáticas, la conjetura de Mertens fue una conjetura según la cual la función de Mertens M(n) estaría por √n. Fue planteada por Franz Mertens en 1897 y se demostró que era falsa en 1985. La conjetura, de haberse demostrado cierta, habría implicado la veracidad de la hipótesis de Riemann. (es) In mathematics, the Mertens conjecture is the statement that the Mertens function is bounded by . 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 Stieltjes), and again in print by Franz Mertens, and disproved by Andrew Odlyzko and Herman te Riele.It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor. (en) In de wiskunde is het vermoeden van Mertens een bewering over het asymptotisch gedrag van de mertensfunctie. Het vermoeden is genoemd naar Franz Mertens, die in 1897 zijn vermoeden uitsprak. In 1985 werd dit vermoeden echter weerlegd. Als het vermoeden van Mertens waar zou zijn geweest, zou daarmee ook de riemann-hypothese zijn bewezen. (nl) Mertens förmodan kallas en speciell olikhet inom talteorin som förmodades vara sann 1897 av Franz Mertens. Om M(n) är Mertensfunktionen lyder förmodan: Mertens förmodan bevisades vara falsk av Andrew Odlyzko och Herman te Riele 1985, men deras bevis var inte konstruktivt. Än idag vet man inte för vilket lägsta n olikheten är falsk. Mertens förmodan var viktig eftersom om den vore sann skulle därav följa att Riemannhypotesen vore sann. (sv) 梅滕斯猜想是数论中的一个猜想，是有關數論中梅滕斯函数上下界的猜想，由汤姆斯·斯蒂尔吉斯在一封于1885年写给夏尔·埃尔米特与弗朗茨·梅滕斯（Franz Mertens）的信中提出。这一猜想如果成立的话可以推出黎曼猜想，不过已被与于1985年证否。 (zh) En théorie des nombres, si nous définissons la fonction de Mertens ainsi : étant la fonction de Möbius, alors la conjecture de Mertens énonce que Stieltjes prétendit en 1885 que M(n)⁄√n était compris entre deux bornes constantes, qui selon lui pouvaient être –1 et 1. Mertens à son tour publia un article en 1897 affirmant, calcul de M(104) à l'appui, que l'inégalité \|M(n)\| < √n lui semblait très probable pour tout n > 1. Or toute inégalité de la forme \|M(n)\| < c√n, c étant un réel positif, implique l'hypothèse de Riemann. Plus précisément, l'hypothèse de Riemann est équivalente à : (fr)
foaf:depiction
dcterms:subject	Disproved conjectures Analytic number theory
Wikipage page ID	379847 (xsd:integer)
Wikipage revision ID	1117895460 (xsd:integer)
Link from a Wikipage to another Wikipage	Mertens function Mellin inversion theorem Mellin transform Bounded function Riemann hypothesis Riemann zeta function Mathematics Möbius function Thomas Joannes Stieltjes Andrew Odlyzko Stieltjes integral Law of the iterated logarithm Disproved conjectures Number theory Numberphile Dirichlet series Lenstra–Lenstra–Lovász lattice basis reduction algorithm Astérisque Counterexample Analytic number theory Charles Hermite Herman te Riele List of YouTubers Springer-Verlag Journal für die reine und angewandte Mathematik
Link from a Wikipage to an external page	http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf http://www.expmath.org/expmath/volumes/13/13.4/Kotnik.pdf https://ghostarchive.org/varchive/youtube/20211221/uvMGZb0Suyc https://web.archive.org/web/20070403143340/http:/expmath.org/expmath/volumes/13/13.4/Kotnik.pdf https://www.youtube.com/watch%3Fv=uvMGZb0Suyc http://www.numdam.org/article/AST_1987__147-148__325_0.pdf
sameAs	Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture Mertens conjecture
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thumbnail	wiki-commons:Special:FilePath/Congettura_Mertens.png?width=300
authorlink	Franz Mertens (en) Andrew Odlyzko (en) Herman te Riele (en)
first	Franz (en) Herman (en) Andrew (en)

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