Talk:Mertens conjecture

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The article says "In 1985, Andrew Odlyzko and Herman te Riele conditionally proved the Mertens conjecture false". But I don't think their result was conditional, was it? They disproved the conjecture without any assumptions? If so that sentence should be amended. (Sam Hopkins) — Preceding unsigned comment added by 209.6.53.33 (talk) 20:35, 25 April 2016 (UTC)

Who did the search for counterexamples up to 10^14 and is the result published somewhere? (Akruppa)

There is an interesting discussion of the Mertens conjecture in Talk:M&ouml;bius_function. Perhaps some of that material could be moved here.

Cleanup:

Dates for S. are inconsistent. Also RH has not been proven false. This page is inaccurate.

--FvdP 17:51, 14 Jan 2005 (UTC)
 * the article does not claim that RH has been proven false. Read more carefully.
 * what exactly, precisely is inconsistent with S. dates ?

Tag for context
I have tagged the article to be cleanup for context. Right now, it is very hard for readers without high level of math background to get a sense of what this is about. Perhaps give a 1 paragraph summary for the layman as introduction in the beginning of the article??

Is the first Sigma missing the upper bound of summation?
It just appears as though there is nothing there. Is this a mistake, or is there something that I'm not understanding about its context in the article? -- Ķĩřβȳ ♥  Ťįɱé  Ø  07:46, 14 August 2006 (UTC)

No more a conjecture
A conjecture is an open question. And this one is closed. Not a conjecture. High time to create a new category. 83.199.53.61 22:30, 1 November 2006 (UTC)

NO IT ISNT. YOU MUST GIVE AT LEAST ONE SPECIFIC VALUE OF N FOR WHICH IT FAILS. The counter proof is via 'probablilities' and so it is not a counter proof. THe most basic requirement of stats is that samples are truly independent. I suspect the existence of close primes breaks the stats requirement. If a sequence is truly random, then it does not have any relationship to either the Moebius or Mertens functions. You must prove the Moebius sequence is truly random, but you won't. First you need a formal definition of 'random' as appropriate for number theory. In this you will fail. Strange relationships can be found between irrationals and transcendental numbers. Formulae may agree to 20 to 30 figures before they diverge. This is no way establishes any sort of poof of a link between them. The Stieltjes paper is quoted as 1885, then as 1985. Which one is it? Mathematics is not worth doing unless you get it exactly right. "Probably wrong" is not good enough. That is the nature of the subject.220.244.245.17 (talk) 04:14, 4 December 2014 (UTC)


 * You are both wrong: On one hand, it has been disproved. Although we don't know a specific counter-example, it has been proved that, e.g., lim sup m(n) > 1, which is rigorously disproving the conjecture (i.e., proving that it is false). And why not create a new category for disproved conjectures. But in cases like this, the statement may keep "conjecture" in its name, because it is well-known by this name. It's a disproved conjecture, so it's a conjecture. Maybe we can establish "Disproved Merten's conjecture" as a convenient new name for future generations. :-) PS: Conjecture says that disproved conjectures are to be called "false conjectures", maybe the 1st phrase of the article could be edited n that sense. &mdash; MFH:Talk 09:43, 15 February 2017 (UTC)
 * With regard to the dates, 1885 was the date of Stieltjes' letter to Hermite, while 1985 (coincidentally 100 years later) was when Odlyzko & te Riele's disproof was published. I don't see any error in these dates in the version current in December 2014. — Pingkudimmi 15:18, 30 April 2020 (UTC)

We could also give the converse the name The Odlyzko - te Riele Theorem. Grassynoel (talk) 11:14, 19 May 2022 (UTC)

Bounds on counter examples
There seems to be disagreement about the upper bound 3.21*10^64 or exp(3.21*10^64). Does somebody have access to a peer reviewed paper? (Not just claims about what such a paper says) Where is the lower bound 10^14, and upper bound 1.59*10^40 or exp(1.59×10^40) from? PrimeHunter 23:49, 31 January 2007 (UTC)
 * [NB: This has been corrected since then on MathWorld. &mdash; MFH:Talk 09:45, 15 February 2017 (UTC)]
 * MathWorld is wrong. Read "Further systematic computations on the summatory function of the Möbius function" by Kotnik and van de Lune. They refer to Pintz for the exp(3.21*10^64) bound. 10^40 was to good to be true :-) // Wellparp 08:13, 1 February 2007 (UTC)


 * See also this talk by te Riele [www.math.tu-berlin.de/~kant/ants/Proceedings/te_riele/te_riele_talk.pdf] // Wellparp 08:17, 1 February 2007 (UTC)


 * Thanks for the link which confirms the 3 bounds the article mentions. I have added the link in the references. PrimeHunter 13:44, 1 February 2007 (UTC)

The integral after the Mellin Inversion Theorem is wrong
The integral after the application of the Mellin Inversion Theorem seems wrong.

The integral is from "sigma-is" to "sigma+is", but the term inside the integral is of the form f(s)ds. Presumably this should be sigma-ix and sigma+ix?

Thomaso (talk) 16:48, 2 January 2008 (UTC)


 * Note that $$x$$ also appears under integral. I think the integral should be taken over the line $$\Re s = \sigma$$ (i.e. from $$\sigma - i \infty$$ to $$\sigma + i \infty$$).
 * 83.6.63.59 (talk) 22:24, 7 February 2008 (UTC)

Statement of conjecture is not accurate.
I have found two counterexamples to the conjecture as stated here.

n=0 gives a counterexample because M(0)=0. O is not less than 0. n=1 gives a counterexample because M(1)=1. 1 is not less than 1.

Is the remedy simply to say that the inequality was only meant to hold for all n>1? Or perhaps it was meant to say |M(n)|<=sqrt(n) for all n>=0? I really want to know exactly what statement has supposedly been disproved! —Preceding unsigned comment added by Latest Incarnation (talk • contribs) 23:23, 27 May 2009 (UTC)


 * I have added "for all n > 1". This is what sources say. PrimeHunter (talk) 23:50, 27 May 2009 (UTC)

There is apparently a related problem with the extreme known negative value found by Kuznetsov. Since M(5) = -2, m(2) = -2/sqrt(5), about -0.8944. Kuznetsov's value might be the best for n > 13. 209.179.95.97 (talk) 20:19, 20 March 2018 (UTC)

Common description
I realize a description understandable by non-math majors may be difficult, but would be appreciated as I'm not clear on how to parse the two equations provided. —Preceding unsigned comment added by Kolano (talk • contribs) 02:29, 6 March 2010 (UTC)

Erroneous redirect Gonek's conjecture
The article mentions a conjecture from Gonek, but I believe that "Gonek's conjecture" nowadays usually refers to a different conjecture that implies $$\forall\epsilon>0:g_n << p_n^{\epsilon}$$, stated in S. M. Gonek: An explicit formula of Landau and its applications to the theory of the zeta function, Contemp. Math 143 (1993), 395–413. [ MR:1210528. So the redirect Gonek's conjecture to this page is somewhat erroneous. &mdash; MFH:Talk 09:55, 15 February 2017 (UTC)

External links modified (January 2018)
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Reference for statement on Riemann Hypothesis
The page says "and valid for 1/2 < σ < 2 on the Riemann hypothesis". There's no reference for this statement, nor any explanation why should this be so. — Preceding unsigned comment added by 95.168.120.132 (talk) 11:52, 23 June 2018 (UTC)

Largest known value of m(n)
In "H. Cohen. Arithmétique et informatique. Astérisque, 61:57–61, 1979." it's claimed that $M(n)$ for $n$ up to $7.8 × 109$ is computed, but the maximum value is not explicitly mentioned. That might be where the claim comes from.

137.132.214.8 (talk) 13:05, 8 May 2022 (UTC)