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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]]]] | |||
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}}. | |||
It is a striking example of a mathematical proof contradicting a large amount of computational evidence in favor of a conjecture. | |||
== Definition == | |||
In [[number theory]], if we define the [[Mertens function]] as | |||
:<math>M(n) = \sum_{1\le k \le n} \mu(k)</math> | |||
where μ(k) is the [[Möbius function]], then the '''Mertens conjecture''' is that for all ''n'' > 1, | |||
:<math>\left| M(n) \right| < \sqrt { n }.\,</math> | |||
== Disproof of the conjecture == | |||
[[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>.) | |||
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 | |||
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> | |||
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> | |||
== 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>\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 ''M'' in terms of | |||
1/ζ 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 1 < σ < 2, and valid for 1/2 < σ < 2 on the Riemann hypothesis. | |||
From this, the Mellin transform integral must be convergent, and hence | |||
''M''(''x'') must be ''O''(''x''<sup>e</sup>) for every exponent ''e'' greater than | |||
1/2. From this it follows that | |||
:<math>M(x) = O(x^{\frac12+\epsilon})</math> | |||
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 | |||
:<math>M(x) = O(x^\frac12)</math>. | |||
==References== | |||
<references /> | |||
* {{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 }} | |||
* 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 | |||
* F. Mertens (1897), "Über eine zahlentheoretische Funktion", ''Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, Abteilung 2a'', '''106''', pp. 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 | 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 }} | |||
* J. Pintz (1987), "An effective disproof of the Mertens conjecture", ''Astérisque'' '''147-148''', pp. 325–333. | |||
* {{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}} | |||
[[Category:Number theory]] | |||
[[Category:Disproved conjectures]] |
Revision as of 02:01, 21 January 2014
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 Hermite (reprinted in Template:Harvnb) and Template:Harvs, and disproved by Template:Harvtxt. It is a striking example of a mathematical proof contradicting a large amount of computational evidence in favor of a conjecture.
Definition
In number theory, if we define the Mertens function as
where μ(k) is the Möbius function, then the Mertens conjecture is that for all n > 1,
Disproof of the conjecture
Stieltjes claimed in 1885 to have proven a weaker result, namely that was bounded, but did not publish a proof.[1] (In terms of , the Mertens conjecture is that .)
In 1985, Andrew Odlyzko and Herman te Riele conditionally proved the Mertens conjecture false: indeed, and .[2][3] It was later shown that the first counterexample appears below exp(3.21Template:E) (Pintz 1987) but above 1014 (Kotnik and Van de Lune 2004). The upper bound has since been lowered to exp(1.59Template:E) (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 (Template:As of). 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 of the Riemann zeta function.[4]
In 1979 Cohen and Dress found the largest known value of 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.[5]
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,
valid in the region . We can rewrite this as a Stieltjes integral
and after integrating by parts, obtain the reciprocal of the zeta function as a Mellin transform
Using the Mellin inversion theorem we now can express M in terms of 1/ζ as
which is valid for 1 < σ < 2, and valid for 1/2 < σ < 2 on the Riemann hypothesis. From this, the Mellin transform integral must be convergent, and hence M(x) must be O(xe) for every exponent e greater than 1/2. From this it follows that
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
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Odlyzko & te Riele (1985)
- ↑ Sandor et al (2006) pp.188–189
- ↑ Template:Cite web
- ↑ Kotnik & te Riele (2006)
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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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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - J. Pintz (1987), "An effective disproof of the Mertens conjecture", Astérisque 147-148, pp. 325–333.
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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