Talk:Riemann hypothesis

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Untitled[edit]

Big O notation[edit]

The article uses a bit of Big O notation when describing the implications of the hypothesis. However, it uses a calligraphic font for the O, which is not used at all, or even noted, in the Big O notation article. This is quite confusing. I'm not sure that there's not a good reason for this, so I won't change it, but if there is a reason it should be noted (perhaps in the Big O article). Otherwise, it should be changed. 75.228.48.146 (talk) 08:45, 3 February 2010 (UTC)

Some people seem to think that these god-awful calligraphic O's look better (don't ask me why). This is not standard notation, so I changed it to normal O. — Emil J. 11:56, 3 February 2010 (UTC)

"Many Propositions"[edit]

The phrase "many propositions relying on the truth of the Riemann Hypothesis" (most likely paraphrased) near the beginning of the section discussing the repercussions of the truth of the Hypothesis should be explained more. What are these propositions? Yes, I understand some of them are discussed in the section, but aren't there others? "Many" would have more. What are these others? —Preceding unsigned comment added by 174.70.46.165 (talk) 02:11, 19 March 2010 (UTC)

"the true importance"[edit]

The article says "It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta-function, which accounts for the true importance of the Riemann hypothesis in mathematics".--I'm not doubting it, but things like that should be backed with a reliable reference. Jakob.scholbach (talk) 09:41, 23 April 2010 (UTC)

External link for Professor Braun[edit]

Monday, 4th October 2010

I have been attempting to add an external link to Peter Braun's website on the Riemann Hypothesis and twin prime problems, however it is continually removed. I am curious as to why this is as I do not believe it is a violation of any of wikipedia's policies and is a valid addition to the site.

Regards Harley. —Preceding unsigned comment added by Harleyjamesmunro (talkcontribs) 07:03, 4 October 2010 (UTC)

This has been removed several times because it has clear original research issues. People often add self-published web papers to Wikipedia mathematics articles, but this is against policy unless the material has been peer reviewed and published in a recognized academic journal.--♦IanMacM♦ (talk to me) 08:11, 4 October 2010 (UTC)

Hello again, i hope you recieve this, as I'm unclear how to talk to website editors on wikipedia. In regards to the website of Peter Braun (my grandfather), i was wondering if you might be able to email him with why you won't allow me to link his website to the page on The Riemann Hypothesis as he has more information on the website then i do, i simply do the webwork. Thank you very much, his address is [redacted]. Regards Harley —Preceding unsigned comment added by Harleyjamesmunro (talkcontribs) 09:30, 13 October 2010 (UTC)

Leaving email addresses like that around is a bad idea; spambots have access to Wikipedia too. In any case, you can email him with why we won't keep your weblink; the reason has been made clear by IanMacM just above your post. Gavia immer (talk) 17:28, 13 October 2010 (UTC)

Lev Pustyl'nikov insertion[edit]

The following text and reference has been inserted; I'm not sure that it's appropriate. I put it here so it could be discussed (and viewed) regardless of the current state of the article.

New results in the theory of classical Riemann zeta function associated with Riemann hypothesis were obtained by L. D. Pustyl'nikov (see Pustyl'nikov (2008) and references therein). The results can be devided into two groups. The results related to the first group are associated with construction of an operator acting in a Hilbert space such that the Riemann hypothesis is equivalent to the problem of the existence of an eigenvector with the eigenvalue -1 for this operator. It is also constructed a dynamical system which turns out to be related to the Riemann hypothesis in the following way: for each complex zero of the zeta function not lying on the critical line, there is a periodical trajectory of order two having a special form. The results related to the second group are associated with the the Riemann \xi (s)-function and its derivatives at the point s = \frac{1}{2}. It is proved that if at least one even derivative of the function \xi (s) at the point s = \frac{1}{2} is not positive the Riemann hypothesis on the zeros of the classical zeta function \zeta(s) would be false. However, it was also proved that all the even derivatives at the point s = \frac{1}{2} are strictly positive and, moreover, the asymptotics for the values of the even derivatives at the same point as the order of the derivative tends to infinity was found. These results permitted to show that the Riemann hypothesis does not hold for an arbitrary sharp approximation of \zeta (s) satisfying the same functional equation and having the same real zeros as the function \zeta (s).
  • Pustyl'nikov, Lev (2008), New results in the theory of the classical Riemann zeta function., Friedr. Vieweg & Sohn Verlag, pp. 187–192  Unknown parameter |books= ignored (help); |first2= missing |last2= in Authors list (help)

CRGreathouse (t | c) 19:13, 12 October 2010 (UTC)

The use who created it is, judging by the name, Pustyl'nikov himself, and his only edits have been to add citations to his own work. That seems inappropriate to me regardless of relevance or significance. —David Eppstein (talk) 19:36, 12 October 2010 (UTC)
While this is clearly a WP:COI, I think we should simply judge the relevance (that is, quality) of the inserted reference. I currently don't have MathSciNet or Zentralblatt here, but could someone check whether this paper has been cited by secondary sources? Jakob.scholbach (talk) 19:42, 12 October 2010 (UTC)
I don't think I have ZB access, but it has no cites in Google scholar and the only reference to it from anything else in MathSciNet is the review of the volume it appears in, which lists it as part of the table of contents. I did find two papers by other authors in MathSciNet that cite P's other papers (MR 2225494 and MR 2285583) and one review that cites one of his papers (MR 2478268). —David Eppstein (talk) 20:14, 12 October 2010 (UTC)
The content added is taken almost verbatim from the paper's abstract (see [1]), so even apart from these problems including this text violates copyright. I've removed it again. Hut 8.5 20:24, 12 October 2010 (UTC)
If the author owns the copyright and posted it here, then that's not a violation. CRGreathouse (t | c) 23:00, 12 October 2010 (UTC)
The article is from a book that claims to be copyright Friedr. Vieweg & Sohn Verlag, "All Rights Reserved". —David Eppstein (talk) 23:12, 12 October 2010 (UTC)
(edit conflict) If he does, and has, yes; but we would need OTRS evidence of that. At present, his work is published in a compilation, and we have no evidence of its copyright status in respect of that work. If he's prepared to release it free of copyright, it should be on Wikisource, and may then be cited here. Until then, we must treat is as being subject to copyright. Rodhullandemu 23:15, 12 October 2010 (UTC)
I was suggesting that (by posting here) he was releasing that portion of the text, not that he'd release the whole text. CRGreathouse (t | c) 00:55, 13 October 2010 (UTC)

How to prove The Riemann Hypothesis(Published Again)[edit]

  • How to prove the Riemann Hypothesis

My paper "how to prove the Riemann Hypothesis" was published in the web Journal"General Science Journal" on March 18th 2005.The address of the Journal is www.wbabin.net.It was published again in the Journal Spacetime&Substance,No.1,2006,P.1.Also it is published in the Proceedings of PIRT-CMS-2007 Kolkata .Now it is published again in The Proceedings of The World Congress on Engineering and Computer Science 2010:WCECS 2010,October 20-22,2010,San Francisco,USA Vol I pp.149-154.Fayez Fok Al Adeh —Preceding unsigned comment added by 88.86.31.173 (talk) 16:34, 16 November 2010 (UTC)

See the talk page archive ad nauseam. This is not recognized by reliable sources.--♦IanMacM♦ (talk to me) 16:41, 16 November 2010 (UTC)
I'm curious, why would you (or anyone) publish in General Science Journal? Are you trying to disseminate information? (Why there, then, since no one takes it seriously?) Are you trying to impress someone? Are you putting this on a resume or a CV?
Also, survey-type question: Do you buy awards from American Biographical Institute or similar companies?
CRGreathouse (t | c) 17:24, 16 November 2010 (UTC)

Please do not react too much "mathematical".[edit]

I certainly can not be this guy pictured above himself, and so I do not know why Wiki was started up. But I do not know, I think I could contribute to this article, to promote it to once again a "good" article by resolving the Riemann Hypothesis. Of course, my edit here is not to disseminate the proof. Well, the proof is correct, and was published (they have their own copyright policy, and they say "published") in arXiv already; it is already a verifiable knowledge.

Have you ever met a guy who stated it so clearly? I think you have met with guys who bravely constructed their own proofs of RH but unfortunately were not correct. But no one who said what I said about Wiki and a proof of RH above, I guess.

Trying to be a polite human being one more time, please keep my edit as permanent.

Should some disproof against my argument occur, then it would be quite natural that the edit be revised again.

Finally, I never have the intention to make a sudden crush against this thread. I am always welcome with any discussion.

But please be polite. Please behave like people who love to know knowledge of any kind. H. Shinya Takemehomecr (talk) 10:56, 17 November 2010 (UTC)

Wikipedia cannot publish original research. This article has long been a popular target for people who want to do this, but the correct route is to publish the material in a peer reviewed academic journal.--♦IanMacM♦ (talk to me) 11:05, 17 November 2010 (UTC)

I also took the time to think about this no-original-research rule. But, if another person edits about my content, then is this rule still applicable to this guy? This guy is now editing about my result, and it is not his own original result.

One exhausting way to overcome this problem is quite simply, verify it. It is a verifiable result. As long as there are many people like you who are very eager to be a sort of moderators, this exhausting ways should work, I believe. I think insisting on peer-reviewed journals are too much mathematical. Takemehomecr (talk) 11:24, 17 November 2010 (UTC)

By the way, I will not attempt to paste the code there anymore. But please give me an agreeable reply to the question above. Takemehomecr (talk) 11:27, 17 November 2010 (UTC)

See the previous reply. By the way, congratulations on winning the Clay Mathematics Institute prize of $1 million for solving one of the Millennium Prize Problems. Where is the massive media coverage of this event?--♦IanMacM♦ (talk to me) 11:30, 17 November 2010 (UTC)

I shall not say anymore.

Sorry for taking your time. Takemehomecr (talk) 11:38, 17 November 2010 (UTC)

I'm an eighth grader, and I don't really understand this article AT ALL. It's too technical. So I wouldn't expect anyone (except maybe Stephen Hawking) to get it. I'm still in the dark as to what areas of math the Riemann hypothesis sheds light on. Can we please improve this article so that anyone, including any random kid at my school, can understand it? Cheers, The Doctahedron, 21:14, 30 December 2011 (UTC)

Clarifying "though there are infinitely many exceptions for larger imaginary part"[edit]

This sentence is grammatically incorrect - perhaps it should read "for larger imaginary part*s*".

Even when corrected, it is unclear what "larger" means. Is it:

  • slightly larger than the imaginary parts of the 3 million zeroes without an exception?
  • larger than a particular value? or
  • sufficiently large?

twilsonb (talk) 01:00, 19 March 2011 (UTC)

There are infinitely many exceptions. The first three million zeros (that is, the six million zeros a + bi for which |b| is minimal) are not exceptions. Thus there are infinitely many zeros which are exceptions for which |b| is greater than the imaginary part of any of the first three million zeros. (The three/six thing is because zeros a + bi are usually identified with their reflection a - bi.)
So your second and your third are correct, trivially. Because there are only finitely many zeros in a finite part of the critical strip, #1 is wrong for any reasonable interpretation of "slightly" -- you can only pack finitely many exceptions in "slightly larger", assuming it's finite.
CRGreathouse (t | c) 20:34, 19 March 2011 (UTC)
I've changed it to: "although there are infinitely many exceptions to Rosser's rule over the entire zeta function." - I gather from your description that this is true, and expresses what is meant better than using a comparative such as 'larger'. twilsonb (talk) 23:55, 27 March 2011 (UTC)
I find that prose significantly worse, but I'll leave it for now. CRGreathouse (t | c) 16:33, 4 April 2011 (UTC)
Please change it if you can think of something that's clear, concise and accurate. It's hard to express and understand what is meant in just a few words! twilsonb (talk) 13:10, 12 April 2011 (UTC)

Zeta grid[edit]

Wedeniwski's calculations seem to have gone up to a higher number at a later date. —Preceding unsigned comment added by 109.158.80.13 (talk) 17:03, 22 March 2011 (UTC)

The Zeta grid calculations seem to reached the first 100 billion non-trivial zeros above the real axis by the December of 2005,
when the project was closed down. —Preceding unsigned comment added by 93.97.194.200 (talk) 12:05, 23 March 2011 (UTC)
Actually, Zetagrid reached 1129.4 billion nontrivial zeros by 31/10/2005. It may be neccessary to use the archive site. —Preceding unsigned comment added by 93.97.194.200 (talk) 11:21, 25 March 2011 (UTC)
Ironically, the faster program and calculations of Gourdon and Demichel overtook those of Zetagrid. —Preceding unsigned comment added by 93.97.194.200 (talk) 11:34, 25 March 2011 (UTC)

How to Prove The Riemann Hypothesis(Republished anew)[edit]

I have proved The Riemann Hypothesis in a paper entitled: How to Prove The Riemann Hypothesis. My proof is exact. My paper "How to Prove The Riemann Hypothesis" was published in the web Journal"General Science Journal" on March 18th 2005.The link of the Journal is www.wbabin.net(List of Authors:al-Adeh Fayez Fok).It was published again in the Journal Spacetime&Substance,No.1,2006,P.1(hard copy available).Also it is published in the Proceedings of PIRT-CMS-2007 Kolkata(pp.18-28, hard copy available). It is published again in The Proceedings of The World Congress on Engineering and Computer Science 2010:WCECS 2010,October 20-22,2010,San Francisco,USA Vol I pp.149-154(Link www.iaeng.org/publication/WCECS2010/). Now it is republished anew in The Journal of Calcutta Mathematical Society Vol 7,No.1,June 2011. Fayez Fok Al Adeh — Preceding unsigned comment added by 213.178.244.16 (talk) 05:40, 2 June 2011 (UTC)

Why didn't you get the $1 million prize? Looks impressive to me. — Preceding unsigned comment added by 98.169.179.61 (talk) 10:35, 16 July 2011 (UTC)

Agreed, it will go in the article after you have won the Clay Mathematics Institute prize. Fayez Fok Al Adeh has been touting this on the talk page since at least 2005.--♦IanMacM♦ (talk to me) 15:27, 16 July 2011 (UTC)
To win the Clay prize you need to get published in a peer-reviewed journal. The 'journals' you listed aren't peer-reviewed. Once it is you can ask on this Talk page to have someone add it for you. (You couldn't just add it yourself because of WP:COI.) CRGreathouse (t | c) 00:59, 17 July 2011 (UTC)

How to Prove The Riemann Hypothesis(Note)[edit]

Most of the Journals in which my paper:(How to Prove The Riemann Hypothesis)is published are peer-reviewed.Fayez Fok Al Adeh are peer-reviewed.Fayez Fok Al Adeh — Preceding unsigned comment added by 213.178.244.16 (talk) 12:19, 18 July 2011 (UTC)

I have bachelor of science in mathematics and having looked at your proof, I can say it is wrong. The equation (5) of your proof is not true. Counter-example -- if s = 0.1 + I, then zeta(s) using the definition of riemann zeta function does not equal to your equation (5) zeta(s). This collapses your proof. — Preceding unsigned comment added by 98.169.179.61 (talk) 05:03, 19 July 2011 (UTC)

The definition of The Riemann Zeta Function[edit]

For the definition of The Riemann Zeta Function,please refer to the book: The Theory of the Riemann Zeta-Function E.C.Titchmarsh Oxford Science Publications. Fayez Fok Al Adeh — Preceding unsigned comment added by 213.178.244.16 (talk) 13:04, 19 July 2011 (UTC)

"log" is ambiguous[edit]

Hi all,

Where I'm from, people use \log for the base-10 logarithm. This article uses that notation inappropriately (i.e. for the natural logarithm). \log_{10} is better than just \log, which is ambiguous. But the usage of \ln instead may clarify things. I already made this correction in one section of the article and will proceed to rectify the remainder thereof.

Thanks,

The Doctahedron, 21:31, 30 December 2011 (UTC)

Gasp! No, no, no. Tradition requires \log- even unto the last breath. Indeed, what sound does a drowning analytic number theorist make? Answer: \log\log\log...

Regards... — Preceding unsigned comment added by 67.85.12.12 (talk) 22:46, 31 December 2011 (UTC)

Excluded Middle[edit]

I'd like to add a (sub)section on proofs that go

"assume the RH is true. (proof of result)"

"assume the RH is false. (different proof of same result)"

"therefore, (result)"

Littlewood's original proof of Littlewood's Theorem (the difference \pi(x)-\operatorname{Li}(x) changes sign an infinite number of times) is like this. (Proof is in Ingham or Landau)

The article on Euler's totient contains

In fact, more is true.[1][2]


\varphi(n) > \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}} 
      for n > 2, and

\varphi(n) < \frac  {n} {e^{ \gamma}\log \log n} 
                      for infinitely many n.

Concerning the second inequality, Ribenboim says "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption."[3]

Questions:

I know there are more examples of this but I can't think of any. (class number?)

Where in the article should this go?

Thanks

Virginia-American (talk) 13:52, 9 January 2012 (UTC)

Another example: As you surmised parenthetically above, the Gauss class number conjecture was first proved in this manner (using the Generalized Riemann hypothesis). This is mentioned at
  • Ireland, K.; Rosen, M. (1993). A Classical Introduction to Modern Number Theory. New York, New York: Springer-Verlag. p. 359. ISBN 038797329X. 
Myasuda (talk) 14:27, 9 January 2012 (UTC)
Thanks, Virginia-American (talk) 17:50, 9 January 2012 (UTC)
References
  1. ^ Bach & Shallit, thm. 8.8.7
  2. ^ Ribenboim, p.320
  3. ^ Ribenboim, p. 320

Unconditional proof[edit]

The phrase “unconditional proof” is used at a number of points in this article in such a way as to imply a special meaning or significance for the phrase — beyond just a simple “proof”. (The phrase also appears in a seemingly similar manner in the Natural proof article.) Not being versed in advanced mathematics, it’s not clear to me what the full meaning of the phrase might be in this context. That suggests it may be a suitable subject for an article to provide a clear definition. So, am I right in thinking there may be a particular meaning for “unconditional proof” in mathematics (and perhaps in other fields involving formal theories and conjectures)? Or, am I just extending some of my confusion regarding the subject of the article to this particular application of language? Thanks for any clarification. —GrantNeufeld (talk) 00:30, 3 March 2012 (UTC)

Commenting as a layman, it seems that "conditional proof" has a specific meaning, namely that the proof is built upon another (as yet unproven) hypothesis – thus, the conditional proof yields a theorem only once that hypothesis is proven. In the article, an example is where the (unproven) generalized Riemann hypothesis is used as a basis for conditional proofs of other results. Those results would then automatically become theorems if the generalized Riemann hypothesis was to be proven. "Unconditional proof" is merely to emphasise that there is no hypothesis upon which the proof rests. — Quondum 06:29, 5 March 2012 (UTC)

Update the web site: http://en.wikipedia.org/wiki/Riemann_hypothesis[edit]

The web site http://en.wikipedia.org/wiki/Riemann_hypothesis must be updated,because of the following:

           How to Prove The Riemann Hypothesis 
                             Fayez Fok Al Adeh 
                              hayfa@scs-net.org  

I have proved The Riemann Hypothesis in a paper entitled: How to Prove The Riemann Hypothesis. My proof is exact. My paper "How to Prove The Riemann Hypothesis" was published in the web Journal"General Science Journal" on March 18th 2005.The link of the Journal is (http://www.gsjournal.net/aladeh/riemann.pdf). It was published again in the Journal Spacetime&Substance,No.1,2006,P.1(hard copy available). Also it is published in the Proceedings of PIRT-CMS-2007 Kolkata(pp.18-28, hard copy available). It is published again in The Proceedings of The World Congress on Engineering and Computer Science 2010:WCECS 2010,October 20-22,2010,San Francisco,USA Vol I pp.149-154(Link http://www.iaeng.org/publication/WCECS2010/). Now it is republished anew in The Journal of Calcutta Mathematical Society Vol 7,No.1,June 2011(pp.1-14). — Preceding unsigned comment added by 213.178.244.16 (talkcontribs) 12:31, 11 March 2012‎ (UTC)

I do not trust that one managed to prove this heavily researched conjecture, but failed to acquire some typography and punctuation usage for his printed article. Incnis Mrsi (talk) 12:45, 11 March 2012 (UTC)
There are several sections above in this page which discuss on Fayez Fok Al Adeh' alleged result. One of the posts asserts it is wrong. The others asserts that it is not peer reviewed. In both cases it can not been included in Wikipedia. D.Lazard (talk) 13:42, 11 March 2012 (UTC)
A paper published in General Science Journal would not be considered a reliable source. There seem to be no WorldCat holdings for this journal, and it doesn't pass the smell test. For instance, a significant proportion of the articles they publish claim to refute Einstein's theory of relativity. Similarly, I can only find one WorldCat holding of the "Journal of Calcutta Mathematical Society". Even the journal's own website only lists articles for a single year. So this also seems questionable to me. Finally, it is our policy not to reference primary sources (such as your paper), particularly for such extravagant claims. Instead, we must find multiple reliable secondary sources in the peer reviewed mathematics literature (in decent journals) that affirm or refute your proof in a substantial way. There are dozens of false proofs of the Riemann hypothesis published each year, but we should only discuss those that have received substantial attention. It's not our job here to read through all of these proofs to provide refutations or to see if somehow the mathematical community missed the one true proof. That has to happen in secondary sources. Sławomir Biały (talk) 13:57, 11 March 2012 (UTC)

Please review the journals in which my paper"How to Prove The Riemann Hypothesis "was published[edit]

It is easy to review The Proceedings of The World Congress on Engineering and Computer Science 2010:WCECS 2010,October 20-22,2010,San Francisco,USA Vol I pp.149-154(Link http://www.iaeng.org/publication/WCECS2010/and The Journal of Calcutta Mathematical Society Vol 7,No.1,June 2011(pp.1-14).in which my paper was published.They are available at requestFayez Fok Al Adeh88.86.31.175 (talk) 12:47, 12 March 2012 (UTC)

See my reply above. Don't start new threads on the same topic while one is already going on. It is considered to be disruptive. Sławomir Biały (talk) 13:51, 12 March 2012 (UTC)
Fayez Fok Al Adeh has raised this issue many times before (see talk page archive) and should know by now that Wikipedia articles cannot comment on original research. Realistically, until the prize on offer by the Clay Mathematics Institute is won, there is little point in claiming that the problem is solved.--♦IanMacM♦ (talk to me) 15:30, 12 March 2012 (UTC)
Interestingly, the very same journal in which Fayez Fok Al Adeh "published" his ground-breaking work has just published a paper which asserts that the RH can neither be proved not disproved. What is a follower of non-peer-reviewed journals to believe? ubiquity (talk) 19:37, 30 January 2013 (UTC)
Both, of course. A belief cannot be narrow-mindedly suppressed by a mere contradiction (especially if it generates revenue to the publisher).—Emil J. 19:53, 30 January 2013 (UTC)

Mistake[edit]

Note that the second equation in the paragraph entitled "History" is wrong. Mu(4)=0. — Preceding unsigned comment added by 78.105.0.33 (talk) 12:31, 16 June 2012 (UTC)

Thanks, fixed. Virginia-American (talk) 16:49, 16 June 2012 (UTC)

dubious venue[edit]

i added the article link involving the Hamiltonian for Riemann Zeros, do you mean that Hindawi is not a reliable source ?? thanks, in the story said something about -dubious venue- or similar i do not understand it. — Preceding unsigned comment added by Karl-H (talkcontribs) 17:49, 26 October 2012 (UTC)

Baseless Riemann Hypothesis[edit]

Lakshan Bandara suggested that Riemann Hypothesis is baseless, where it can neither be true nor false. http://www.gsjournal.net/Science-Journals/Essays/View/4491 — Preceding unsigned comment added by 112.134.176.24 (talk) 19:15, 30 January 2013 (UTC)

The suggestion seems to be, at best, speculative, and a mention would be according it undue weight until it is discussed in independent sources. Actually, I think it's either meaningless or false, but the question is what, if anything, independent reliable sources say about it. Deltahedron (talk) 19:23, 30 January 2013 (UTC)

Link to Wu-Sprung Potential[edit]

I have reverted the See Also link to Wu-Sprung Potential since it is not clear that this article is sufficiently relevant. (I assume Wu–Sprung potential was intended.) Deltahedron (talk) 18:17, 19 February 2013 (UTC)

Riemann hypothesis[edit]

86.15.102.53 (talk) 06:57, 22 June 2013 (UTC) :) Just read this bit on your Riemann hypothesis wiki page.

   "In 1923 Hardy and Littlewood showed that the generalized Riemann hypothesis
    implies a weak form of the Goldbach conjecture for odd numbers:
    that every sufficiently large odd number is the sum of three primes,
    though in 1937 Vinogradov gave an unconditional proof.
    In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed that the
    generalized Riemann hypothesis implies that every odd number greater than 5
    is the sum of three primes".
:) I might be being an idiot here but, assuming 7 is the first odd value
   greater than 5 - then the prime 3 must be used twice - 3+3+1=7.
   That being the case the prime 5 can be included: 1+1+3=5.
   And assuming the same prime can be used for ANY prime: 1+1+1=3
Yes, you're allowed to use primes more than once for this (see Goldbach's weak conjecture), but you've made a mistake in assuming that 1 is prime - it isn't. You have to write 7 = 2 + 2 + 3. 5 doesn't work because the only primes less than 5 are 2 and 3, and you can't add three of these to get 5. Hut 8.5 10:27, 22 June 2013 (UTC)

Riemann Hypothesis: An Algebraic Topologic Proof[edit]

A full geometric proof of the Riemann hypothesis has been published on arXiv by A. Prástaro, recasting the problem in the algebraic topology of quantum manifolds, as already published by the same author in some series of works.

See Refs. [1]. In Refs. [2, 3] are also reported some already published papers directly related to [1].
[1] A. Prástaro, The Riemann hypothesis proved. arXiv: 1305.6845[math.GM].
[2] A. Prástaro, Surgery and bordism groups in quantum partial differential equations.I: The quantum Poincaré conjecture, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 502-525. DOI: 10.1016/j.na.2008.11.077. MR2671857(2012b:58057); Zbl 1238.58025.
[3] A. Prástaro, Quantum exotic PDE's. Nonlinear Analysis. Real World Appl. 14(2)(2013), 893-928. DOI: 10.1016/j.nonrwa.2012.04.001. arXiv: 1106.0862[math.AT]. MR2991123.
Actually paper [1] has been submitted to a peer-reviewed mathematical journal, for publication too.
Interested readers are invited to send their comments to agostino.prastaro@uniroma1.it

(87.21.19.37 (talk) 14:29, 3 July 2013 (UTC))

(Agostino.prastaro (talk) 16:53, 3 July 2013 (UTC)) (Agostino.prastaro (talk) 17:04, 3 July 2013 (UTC)) (Agostino.prastaro (talk) 22:08, 27 July 2013 (UTC))

Nothing to do here unless/until it has been reliably published (arXiv doesn't count for that, they only filter for whether it's on-topic, not for whether it's correct). But the listing in math.GM rather than in the more topic-specific category is not promising...that's the catch-all category for crankery. —David Eppstein (talk) 18:00, 3 July 2013 (UTC)
Almost right ! In my opinion the GM arXiv-classification is suitable for the Riemann hypothesis, since it concerns a subject of general interest in Mathematics ...
But you can spend a bit more energy to understand that my proof is correct ... whether, of course, you know Algebraic Topology. This comment should be more appropriate !

(Agostino.prastaro (talk) 18:37, 3 July 2013 (UTC))

(ec) If accurate, and this work is accepted by a peer-reviewed journal, and there is no significant criticism of the author/editor's previous papers, then it might be included in this article. (As an occasional peer-reviewer, I often assume that the referenced papers are legitimate and say what the current author says they do. In the case of potentially-fringe papers, that assumption needs to be checked.) — Arthur Rubin (talk) 18:07, 3 July 2013 (UTC)
Good wishes to Agostino Prástaro but it would WP:UNDUE to mention the preprint unless some independent reliable source suggests that the proof is either correct or at the very least an interesting attempt. Spectral sequence (talk) 18:30, 3 July 2013 (UTC)

Could someone please make it a bit simpler?[edit]

I want to understand what this is about but the wiki page is too mystifying. — Preceding unsigned comment added by Sonicnomad (talkcontribs) 08:17, 22 July 2013 (UTC)

Well, there is simple.wikipedia's version here: http://simple.wikipedia.org/wiki/Riemann_hypothesis
It is, possibly, too simple - it seems like there's a rather large middle ground in which some of the implications of the RH beyond the prime number distribution could be discussed in a less formal sense. Perhaps there should be a detailed summary section of some form that explains why the RH is important in terms that an average well educated individual who is not familiar with the technical jargon and notation used by mathematicians. 24.69.217.16 (talk) 07:16, 2 December 2013 (UTC)

Distribution of Primes[edit]

It should be clearly stated that Riemann's formula to predict the exact number of primes below a given number using the integral of 1/log(x) and the non-trivial zero of the zeta function, is dependent on all those roots lying on x=1/2.

Considering this, it should be explained that if the hypothesis is true the distribution of all primes can be easily determined by using this formula and comparing input magnitudes and any relevant changes in the quantities of primes below adjacent magnitudes.

I'd do it, but every time I try and put up this common knowledge, the info is deleted. — Preceding unsigned comment added by VerticalNexus (talkcontribs) 09:12, 4 August 2013 (UTC)

The connection between the location of the zeros of the Riemann zeta function and the distribution of prime numbers is already very clearly explained in sections 2 and 3.1 of the article.Sapphorain (talk) 09:38, 4 August 2013 (UTC)

Nowhere in either section does it explain 'how' COULD 'easily' determine the distribution of all prime numbers using the non-trivial zeros, and the integral of 1/log(x), if all non-trivial roots of the zeta function lie on x=1/2 in the complex plane.

My main complaint is that it is incredibly easy to determine the distribution of all primes given the validity of the hypothesis, and no where in the article is this method explained.

As if one knows there are 4 primes at or below the magnitudes of 7,8,9, and 10, but 5 primes at or below 11, one knows that 11 is prime. This means that for any prime number, no matter how large, its distribution can be easily known with absolute certainty, as long as the hypothesis is correct. — Preceding unsigned comment added by VerticalNexus (talkcontribs) 10:35, 4 August 2013 (UTC)

Within page 4 of his publication "On the number of prime numbers less than a given quantity," Riemann gives us a method for easily determining the distribution of all prime numbers as long as all of the non trivial roots of the zeta function lie on x=1/2. He states: "...the number of prime numbers that are smaller than x can now be determined." He goes on to say

"Let F(x) be equal to this number when x is not exactly equal to a prime number; but let it be greater by 1/2 when x is a prime number, so that, for any x at which there is a jump in the value in F(x), F(x) = [F(x + 0) + F(x - 0)]/2"

Thus, one can easily determine the distribution of all primes other than the number 2 by simply choosing magnitudes of x which are greater than two and even, and thus not prime, and then comparing the quantities of primes below this even(x) with the quantities of primes of F(x+2),F(x+4),F(x+6),F(x+8)etc...

and continuing until you notice a change in the quantity of primes;that is an increase in f(x); then you note that input magnitude which gave you the increase in f(x) and just subtract 1 to obtain the prime number. VerticalNexus (talk) 12:32, 4 August 2013 (UTC)

VerticalNexus (talk) 12:20, 4 August 2013 (UTC) — Preceding unsigned comment added by VerticalNexus (talkcontribs) 12:16, 4 August 2013 (UTC)

Content is unfathomable to lay people[edit]

The point of Wikipedia, I thought, was to make information accessible to non-specialists. Why is it, then, that pages on mathematics topics in general and this one in particular are so opaque? I doubt that anyone who doesn't already have a mathematics degree could understand this stuff as it is presented. And if you expect that level of understanding of your readers, what's the point of the article? I get the feeling that the kind souls who write and edit this material do not have the correct target audience in mind.

I just feel that Wikipedia is letting its readers down if so much of its material is inaccessible to ordinary people. — Preceding unsigned comment added by 86.158.156.202 (talk) 01:10, 1 September 2013 (UTC)

We have been trying to make it as accessible as possible, but that is not the same as making the whole article accessible to all readers. If we were only allowed to have Wikipedia articles about subjects that could be made easy to understand by everyone, then much of mathematics beyond the high school level would be missing. This is not a modern phenomenon: see Royal Road#A metaphorical “Royal Road” in famous quotations. —David Eppstein (talk) 02:30, 1 September 2013 (UTC)
Richard Feynman was once asked by a journalist describe briefly the work that won him a Nobel Prize, and replied "Listen, buddy, if I could tell you in a minute what I did, it wouldn't be worth a Nobel."[2] Some things in life are not easy to explain in a cut out 'n' keep guide on a breakfast cereal packet, and the Riemann Hypothesis is one of them. Of the famous unsolved mathematical problems, Goldbach's conjecture is the easiest to understand. Many of the Millennium Prize Problems would require an advanced knowledge of mathematics to understand them, let alone solve them.--♦IanMacM♦ (talk to me) 05:53, 1 September 2013 (UTC)
A graph, in theory in four dimensions, would show the first few cases of the non-trivial zeros. — Preceding unsigned comment added by 86.181.10.231 (talk) 11:35, 27 December 2013 (UTC)
You mean like the very first image already in the article? —David Eppstein (talk) 18:20, 27 December 2013 (UTC)
The graph on the top at the right is actually in two dimensions. Also, it does not show the trivial zeros or the pole of the zeta function. — Preceding unsigned comment added by 86.181.10.231 (talk) 11:19, 28 December 2013 (UTC)

How about one sentence on why this matters? 72.208.148.85 (talk) 12:15, 25 February 2014 (UTC)

72.208.148.85 does not say what his word "this" refers to. It might refer to the hypothesis or an easily understood account of it.
Why does the Riemann hypothesis matter? Why is it worthy of an article that only seems to speak to those who already have the knowledge? 72.208.148.85 (talk) 03:20, 22 March 2014 (UTC)

One of Pythagoras's contemporaries asked much the same. — Preceding unsigned comment added by 77.126.93.17 (talk) 12:02, 17 March 2014 (UTC)

Nobody really knows why the Riemann hypothesis matters at the moment. In 1931, Paul Dirac wrote in Quantized Singularities in the Electromagnetic Field "Non-euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have now been found to be very necessary for the description of general facts of the physical world." We would not have iPhones without quantum theory, but the mathematicians who formulated the underlying theories many years earlier did not set out to invent the iPhone. When asked about the use of newly-discovered electricity, Michael Faraday is supposed to have said "Why, sir, there is every probability that you will soon be able to tax it."[3]--♦IanMacM♦ (talk to me) 07:24, 22 March 2014 (UTC)
And when asked why he wanted to climb Mount Everest, George Mallory is supposed to have answered "Because it is there". Sapphorain (talk) 09:09, 22 March 2014 (UTC)

From the lede: "it is considered by some mathematicians to be the most important unresolved problem in pure mathematics" -- and nobody can say why? Can you at least indicate why this shouldn't be added to Articles for Deletion? 72.208.148.85 (talk) 03:32, 23 March 2014 (UTC)

The Clay Institute summary is "A proof ... would shed light on many of the mysteries surrounding the distribution of prime numbers." [4]. Deltahedron (talk) 07:51, 23 March 2014 (UTC)
Thank you for that; I've put your sentence in the article where more people can see it. If I've done so incorrectly, please feel free to improve. 72.208.148.85 (talk) 15:13, 23 March 2014 (UTC)

Did John Nash contribute anything to proving this hypothesis?[edit]

I saw in a documentary film of his life that the stress of his failed attempt to prove it contributed to his development of schizophrenia. Was his aborted work on the hypothesis picked up by any of his colleagues or others?

On MarkAndrewGerads's edit[edit]

Dear MarkAndrewGerads. On Arxiv there are a lot of proofs for R.H.. If you would like to describe them you should be not below Headline but more in suitable other position, for example, in the note or in the under place of the related methods. Otherwise, every attempt to prove R.H. would be described.--Enyokoyama (talk) 03:03, 16 November 2014 (UTC)