Talk:Mertens conjecture

message: 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)
thread_title: ...

message: 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♥ŤįɱéØ 07:46, 14 August 2006 (UTC)
thread_title:  Is the first Sigma missing the upper bound of summation? 

message: 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)
replies: text: 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#Disproof says that disproved conjectures are to be called "false conjectures", maybe the 1st phrase of the article could be edited n that sense. — MFH:Talk 09:43, 15 February 2017 (UTC)

text: 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)
thread_title:  No more a conjecture 

message: 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)
replies: text: [NB: This has been corrected since then on MathWorld. — MFH:Talk 09:45, 15 February 2017 (UTC)

text: 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)

text: 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)
replies: text: 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)
thread_title:  Bounds on counter examples 

message: 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)
replies: text: I have added "for all n > 1". This is what sources say. PrimeHunter (talk) 23:50, 27 May 2009 (UTC)
thread_title:  Statement of conjecture is not accurate. 

message: 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)
thread_title: Common description

message: 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. — MFH:Talk 09:55, 15 February 2017 (UTC)
thread_title:  Erroneous redirect [[Gonek's conjecture]] 

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thread_title:  External links modified (January 2018) 

message: 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)
thread_title:  Reference for statement on Riemann Hypothesis 

message: In "H. Cohen. Arithmétique et informatique. Astérisque, 61:57–61, 1979." it's claimed that  for  up to  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)
thread_title:  Largest known value of m(n)