# Talk:Riemann hypothesis

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## Big O notation

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"

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"

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

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. 17:28, 13 October 2010 (UTC)

## Lev Pustyl'nikov insertion

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 ${\displaystyle \xi (s)}$-function and its derivatives at the point ${\displaystyle s={\frac {1}{2}}}$. It is proved that if at least one even derivative of the function ${\displaystyle \xi (s)}$ at the point ${\displaystyle s={\frac {1}{2}}}$ is not positive the Riemann hypothesis on the zeros of the classical zeta function ${\displaystyle \zeta (s)}$ would be false. However, it was also proved that all the even derivatives at the point ${\displaystyle s={\frac {1}{2}}}$ 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 ${\displaystyle \zeta (s)}$ satisfying the same functional equation and having the same real zeros as the function ${\displaystyle \zeta (s)}$.
• 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)

• 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".

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" 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 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) 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)

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

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

Hi all,

Where I'm from, people use ${\displaystyle \log }$ for the base-10 logarithm. This article uses that notation inappropriately (i.e. for the natural logarithm). ${\displaystyle \log _{10}}$ is better than just ${\displaystyle \log }$, which is ambiguous. But the usage of ${\displaystyle \ln }$ 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 ${\displaystyle \log }$- even unto the last breath. Indeed, what sound does a drowning analytic number theorist make? Answer: ${\displaystyle \log \log \log }$...

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

## Excluded Middle

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 ${\displaystyle \pi (x)-\operatorname {Li} (x)}$ 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]

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

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

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

           How to Prove The Riemann Hypothesis
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

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

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

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

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)

## Riemann hypothesis

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)

## Could someone please make it a bit simpler?

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)
I used to love math but now it's all Greek to me. I looked at Britannica's article and it's a lot more concise and somewhat less daunting: http://www.britannica.com/topic/Riemann-hypothesis Gmporr (talk) 20:01, 17 November 2015 (UTC)

## Distribution of Primes

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

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?

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?

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)

## Dealing with WP:OR and WP:PROMO

I see this page has become something of a crank magnet. Do note that WP:TPO permits/encourages the deletion of promotional material on Talk pages. You do not have to passively let crackpots disrupt the good stuff here. Just aggressively (but politely) revert them and if they put it back, take them to WP:ANI. Choor monster (talk) 11:37, 7 June 2015 (UTC)

## Sarnak official Clay Math statement on RH

The following 9 page document, dated 2004, is what the Clay Math Institute posts by Sarnak on its RH page. There is no mention of de Branges older work, and certainly none of his newer work. What are you referring to? Choor monster (talk) 19:26, 28 July 2015 (UTC)

The paragraph in question is the last full paragraph of page 6 of your link. I got the survey from a different link where it is formatted differently, but apparently with the same text, and with the same paragraph on the bottom of page 19. —David Eppstein (talk) 19:32, 28 July 2015 (UTC)
OK, found it. I am now totally mystified at what your justification is for the new WP material. Sarnak only refers explicitly to the older attempts, he has nothing to say beyond the 2000 refutation. He doesn't even bother citing de Branges. Choor monster (talk) 19:46, 28 July 2015 (UTC)
The version of the article without the text in question implies that de Branges' efforts stopped with his 1992 paper, after his positivity condition was shown not to hold. This is false, and his later attempts are notable despite being somewhat fringe. They are mentioned by Sarnak (in vague terms, but clearly referring to ongoing efforts as of the 2004 date of Sarnak's survey rather than the 1992 paper), and by multiple more recent book sources [5] [6] [7]. —David Eppstein (talk) 20:34, 28 July 2015 (UTC)
The three links you provide only talk about the pre-2004 work of de Branges. Notability is not required, of course, for mere passing references, the issue here is a question of WP:UNDUE. Who, exactly, has commented on de Branges's post-2004 updates? Choor monster (talk) 20:49, 28 July 2015 (UTC)
They all talk about a 2002 preprint which over time has morphed into the preprint discussed here. It is clearly not the 1992 paper. —David Eppstein (talk) 20:56, 28 July 2015 (UTC)
The article can only mention what the RS's discuss, which apparently stops with the 2002 preprint. Until some number theorist thinks it's worth catching up the rest of the world on de Branges's continued attempts, for us to provide the additional information is violating WP:V, WP:OR, WP:UNDUE, WP:NOTNEWS, WP:NOTFORUM, and WP:BLP. That's right: any claims regarding a purported proof of RH is contentious regarding the author.
For the record, the Borwein et al book does not refer to the 2002 preprint, but to earlier work. This despite the fact the book has a 2008 copyright, and a timeline through 2004. Sarnak does not cite any of de Branges, he refers only to a 2000 paper refuting earlier work. Interpreting his words as hinting at the 2002 preprint is already a stretch.
I'm reverting, and if you put it back without an accurate citation (please, don't bother with faking it like these previous 0-for-4 citations) I'm taking it to an admin board. Choor monster (talk) 14:17, 29 July 2015 (UTC)

I think de Branges' attempt is notable. Finding a wording that is mutually acceptable is vastly preferable to outright deletion. 15:28, 29 July 2015 (UTC)

Then blog about it somewhere. No RS, then WP:NOTFORUM. Really, it's that simple. Choor monster (talk) 15:34, 29 July 2015 (UTC)
Yes, de Branges' attempt is notable. De Branges is quite famous for outstanding achievements in mathematics and is not a crank. A more cautious wording for the attempt(s)' description might be necessary, but outright deletion (of Eppstein's contribution) is definitely not an option. Sapphorain (talk) 16:06, 29 July 2015 (UTC)
Well, I think Sarnak is a reliable source, given that Sarnak is one of the world's leading experts on the Riemann hypothesis. Could you state more clearly why you think this source is unreliable? 17:27, 29 July 2015 (UTC)
Please note that I did not object to the previous mention of de Branges's work. That was RSed. I was objecting to the new material, which was not RSed. It's certainly well-known in mathematical circles that he hasn't quit, but WP is not for passing along gossip, no matter how true it is, but for reflecting what has made it into RS.
Yes, [Sarnak 2005] is of course RS. Misquoting it, misrepresenting it, or even making plausible inferences from it, however, is simply not allowed. In particular, it does not refer to anything post-2005. Why is this so confusing? As I spelled out, neither Sarnak, nor the other 3 RS's Eppstein provided above, supported the bit that was put in. It seems Eppstein now agrees, as he has watered it down significantly. Other than improving the accuracy of the reference (its year is 2005, see end of paper), I see nothing to disagree about anymore. Choor monster (talk) 19:38, 29 July 2015 (UTC)
Ok, then let's work on bringing the discussion of de Branges' work more into accord with sources, if the current text of the article does not. Nontrivial discussion of it in reliable sources strongly suggests that some mention of it should be made here. So I'll reiterate that I don't really see outright removal as justified. 14:37, 30 July 2015 (UTC)
It's already been done. I don't see what your point is. Choor monster (talk) 15:10, 30 July 2015 (UTC)

## Sarnak 2005 citation

I changed the Sarnak 2005 citation around to properly reflect it being a previously published piece. However, it should be in the same format as the Bombieri 2000 chapter citation. I have to get back to real-life, however, and will make the changes tomorrow if no one else does. Choor monster (talk) 19:54, 29 July 2015 (UTC)

Do no worry. We shall survive until tomorrow. Enjoy real-life. Sapphorain (talk) 20:43, 29 July 2015 (UTC)
Properly enjoyed, the cite is now updated, along with the harv refs. Choor monster (talk) 15:11, 30 July 2015 (UTC)

## Semi-protected edit request on 17 November 2015

Dr. Opeyemi Enoch has solved the Riemann hypothesis. Source:http://www.academia.edu/1188935/Proof_of_the_Riemann_hypothesis


24.112.113.219 (talk) 07:53, 17 November 2015 (UTC)

Unlikely. And academia.edu is not a reliable source. I know this is in all the Nigeran newspapers, but it looks bogus; see http://www.nairaland.com/2739995/opeyemi-enoch-not-solved-riemann for reasons why. Let's wait unless/until some real mathematicians weigh in. —David Eppstein (talk) 08:12, 17 November 2015 (UTC)

## In the news: Dr. Opeyemi Enoch of Nigeria

There's a news story making the rounds that says that Nigerian math professor Dr. Opeyemi Enoch has solved the Riemann hypothesis. I've noticed that several people have already added it to this page and then had it deleted. Perhaps it would be a good idea to put that template on this article that says e.g. "This page is under semi-protection because its topic has appeared in the news recently." So far, the most reputable source to have reported this story seems to be the BBC World Service. But, there doesn't appear to be any announcement from the Clay Mathematics Institute yet. And none of the news stories have any input from anyone other than Dr. Enoch, like other mathematicians confirming the proof. So it seems like it would probably be a good idea to wait until this story settles down a bit more before actually adding it to the article. --124.197.39.93 (talk) 06:44, 17 November 2015 (UTC)

The point is he didn't post his proof (in any form) yet. The one he uploaded on academia is written by other people. — Preceding unsigned comment added by Williamhchuang (talkcontribs) 10:56, 17 November 2015 (UTC)

Yes, this is really the relevant point. In the recent CNN article, Enoch says 'the mathematical community is behind me,' yet he has declined to show any proof to the mathematical community, or to show a proof anyone at all(except Nina Ringo of the 'vienna conference' see also http://avoidpseudoscience.blogspot.co.uk/2011/12/fake-paper-was-accepted-by-nina-ringos.html). Judging from the first BBC world service article, he incorrectly told the presenter that he had collected the 1 million dollar prize already, which is how the story got started.Subsequent articles by the mainstream press have essentially been retractions of the original articles. Createangelos (talk) 19:44, 19 November 2015 (UTC)
OK now Nina Ringo's web page has a link to her 34 page journal where she says Opeyemi's entire proof is published (along with other articles). Can someone buy this and report on this talk page where the first mistake is pls; this is assuming, one says for the sake of fairness, that there is a mistake. Unfortunately it is not electronic, you have to wait for a 34 page paperback to get delivered probably from Vienna. Nina Ringo herself includes two details which seem to be i) the representation of a complex number as a 2x2 real matrix, and ii) the formula for the characteristic polynomial of a 3x3 matrix. Also a rational function shown which has one pole at z=1 and one real zero at a negative integer and two complex poles of imaginary part 1/2, with the statement that it is 'equivalent to' the zeta function.Createangelos (talk) 21:01, 25 November 2015 (UTC)
That's not how it works. We should not be in the business of finding mistakes in mathematical proofs here, especially not if it requires outside purchases. The proof needs to be accepted by the mathematical community. In this particular case, the whole journal does not appear to be accepted by the mathematical community: it is not listed in MathSciNet, unlike essentially all reputable mathematics journals. —David Eppstein (talk) 19:03, 25 November 2015 (UTC)
It's not that we "should not", but we are flat out not allowed to. Evaluating primary sources is 100% outside our remit. Furthermore, whether it's only available for free, by subscription, or purchase is irrelevant, and is neither weight in favor or against so far as WP rules go. So far as common sense goes, the purchase-only non-electronic form says SCAM, but again, that doesn't really weigh in either. Choor monster (talk) 19:19, 25 November 2015 (UTC)
I think the for-purchase part is relevant, though, under verifiability: print vs online is irrelevant, but material used as sources needs to be something readers can verify, possibly by looking online and possibly by going to a library. It's not reasonable to expect verification to happen for material that can't be accessed in any way except for-purchase. So unless there are library holdings of Ringo's journal (I haven't checked), we have another problem beyond the prohibition on doing original research ourselves. —David Eppstein (talk) 19:31, 25 November 2015 (UTC)
Incorrect. WP:PAYWALL (part of WP:V) is absolutely clear: cost is not an issue when it comes to sourcing. Choor monster (talk) 20:04, 25 November 2015 (UTC)
PAYWALL is about online sources. This is not an online source. It is a self-published print book, by someone with a history of publishing unreviewed or poorly reviewed material, and is apparently unavailable through libraries. —David Eppstein (talk) 20:07, 25 November 2015 (UTC)
Nope, WP:PAYWALL is about printed sources as well. Also, is there a reliable source for this claim of "history"? — Preceding unsigned comment added by 157.181.161.111 (talk) 13:59, 26 November 2015 (UTC)
"a print source may be available only in university libraries or other offline places." It's pretty clear to me that this is referring to conventional sources that can be accessed by researchers with library access, not obscure sources you have to pay directly to the author to verify. 14:07, 26 November 2015 (UTC)
It seems very clear to me: "other offline places" means "other offline places." And over the years, I've come across rare crackpot manuscripts at university libraries. They may have been donated just to get in libraries, or when the expected monetary rewards did not happen, or when one purchaser decided to share. Wait sometime and one of Dr. Enoch's pamphlets will end up at a university library. No, the problem simply is that it's WP:PRIMARY, and as such it can't be used on WP for pretty much anything interesting whatsoever.
As an example of something rarely seen at any library that we regularly accept on WP are scans of rare first edition covers. Libraries used to throw the dust jackets away automatically, so you're mostly confined to shopping, and at high prices. I'm happy to have contributed a few from my private collection. Some are not for sale at the moment; I apparently bought the last copy. Choor monster (talk) 16:04, 26 November 2015 (UTC)
Great. When this print-on-demand manuscript starts being indexed by libraries, then you can claim that it meets WP:V. But until then, there is no reason to believe that this source even exists. 16:37, 26 November 2015 (UTC)
I never claimed it existed. But being indexed by libraries is not a requirement for WP:V either. If you don't like WP's standards, take it up on the WP:V talk page or VPP. You're not going to change anything by complaining about it here. Choor monster (talk) 16:51, 26 November 2015 (UTC)
So, existence is not a prerequisite for verifiability? That would seem to require a radical re-interpretation of the guideline, which says: "verifiability means that anyone using the encyclopedia can check that the information comes from a reliable source." 17:06, 26 November 2015 (UTC)
• Out-of-sequence comment to an IP above: Regarding "history", Eppstein was commenting about what is generally required for a Reliable Source. Those parts of his comments were accurate. The part that it has to be available for free (or close enough) were inaccurate. Choor monster (talk) 17:33, 26 November 2015 (UTC)
I was disapproving the "by someone with a history of publishing unreviewed or poorly reviewed material" part, because the link points to a web page (a blog profile created for this sole purpose) of an unknown person with a bold claim and a couple of claimed emails as "proof" --- now that is one long history of consistently unprofessional publishing, with an official record to prove it --- NOT. — Preceding unsigned comment added by 157.181.161.111 (talk) 16:53, 27 November 2015 (UTC)
You are taking something I said out of context, arguing against a straw man, and making a mess where there was no need of dispute. I certainly do not feel that sources used here are required to be free. But, they must be in some way accessible, and I don't think that being told that you have to buy your own copy of this self-published book because nobody else has a copy counts as accessible. Additionally, I was reacting against the implied sentiment that as Wikipedia editors we should feel duty-bound to pay up for this source so we can use it. There is no obligation, neither for editors to pay out of their own pockets for sources nor for us to pay any attention to this source, which in any case likely fails WP:RS due to its self-published nature. —David Eppstein (talk) 18:05, 26 November 2015 (UTC)
I apologize if you think you were quoted out of context. I still don't see it. Buying your own copy of a self-published book is indeed a means of access. I agree completely with everything else you've stated against Enoch's paper, and why the suggestion that somebody here get cracking and buy a copy was pointless, and if anything, I felt you simply quit too early in enumerating why this "source" is worthless for WP, and I even added to your hit-list. Meanwhile, one of your objections was erroneous, context or no context. Choor monster (talk) 17:36, 27 November 2015 (UTC)
The problem of free access and the issue of cost are, unfortunately, totally irrelevant in this matter. An article claiming any result in mathematics must be accepted by the mathematical community before it is mentioned in an encyclopedia, provided of course the result is sufficiently important. In order to be accepted by the mathematical community the article should be published in a reliable peer-reviewed journal of mathematics. Any reliable peer-reviewed journal of mathematics is reviewed, or at least indexed, either by MathSciNet or by Zentralblatt MATH (usually by both). But the journal which is the object of this discussion is neither reviewed nor indexed by MathSciNet or Zbl. So it is not a reliable journal, and should not be even mentioned in a wikipedia article. Now the fact that neither MathSciNet nor Zbl are free online, and that most good mathematical journals aren't free either, is of course regrettable, but there is simply no alternative acceptable authority on mathematics. Sapphorain (talk) 23:15, 27 November 2015 (UTC)
Then Wikipedia, with its rules for reliable sources, is not an encyclopedia by your definition. Feel free to challenge either the currently existing rules or the The Free Encyclopedia subtitle.

OK I agree we should not buy this. Nina and Opeyemi could earn their million just from the publicity and sales.... But note that the first formula on the top of page 2 of Nina Ringo's publicly available description of the proof https://ninaringo.files.wordpress.com/2015/11/about-the-proof-of-riemann-hypothesis.pdf states a false formula. The left side is presumably the Riemann zeta function since it is labelled \zeta(z), and the right side assuming n and p(j) are numbers, is a rational function. It is known that the Riemann zeta function is not a rational function therefore the statement is false. Also thx for restoring my comment with strikes as quite appropriate.
(PS here is another interesting thing...an article with the same abstract as Opeyemi gave in the Ringo 'conference' actually exists since 2013, here is a link http://www.elixirpublishers.com/articles/1366630964_57A%20(2013)%2014417-14419.pdf although the fonts are unreadable unless one zooms in a lot, and even then pretty unreadable. But it looks more credible than Ringo's description. -- added April 2016: I emailed Opeyemi at the time to ask if the 2013 paper is related to the Riemann Hypothesis. It has been five months and he hasn't replied yet. All that I can find that he's done from the beginning is a) upload a paper of Werner Raab to his Academia page titled 'proof of the Riemann hypothesis', b) give a talk at Ringo's conference with the abstract identical to the 2013 paper, about approximations only, which doesn't prove the Riemann hypothesis c) to go on talk shows and say that Raab is no longer alive, d) to go on talk shows and claim that a Yale PhD student accused him of plagiarizing the (nonexistent) proof, e) to remove the Raab paper from his Academia page and post in its place an announcement stating his page has been hacked and he is not the one who uploaded the Raab paper, f) to post an announcement of a 'speech' at an upcoming Ringo conference to discuss the fact of having proved the Riemann hypothesis at some time in the past, but still with not one detail of what methods might have been involved or where the proof is located).Createangelos (talk) 20:59, 25 November 2015 (UTC)

You're welcome. If you revise quickly, it's no problem, but once it's been commented on by others, you're basically stuck. Another common trick is to simply post your revised version at the bottom of the thread with a boldfaced Revised Comment or other description, preceded by a bullet point. You may also want to introduce four dashes "----" above your comment, it will introduce a separating line.
Not only is the zeta function not rational, there is no rational combination of it and its higher derivatives which is 0, a theorem of Hilbert. A much deeper theorem due to Voronin is that there are no continuous function combinations of zeta and its derivatives that are zero. Precise statements and references, see also Voronin universality theorem. Zeta is one truly difficult beastie to understand. Choor monster (talk) 16:04, 26 November 2015 (UTC)

I think that a small note stating that there is a claimed solution and the matter is under investigation wouldn't be unreasonable. It's rather disorienting to come to this page off the news link and not see even a brief mention. — Preceding unsigned comment added by 101.191.121.218 (talk) 08:32, 17 November 2015 (UTC)

But if we had a "small note" referring to every crackpot's claim to a proof, we would have dozens of small notes, permanently… And this would be rather disorienting. Sapphorain (talk) 09:06, 17 November 2015 (UTC)

I was of course absolutely thrilled to see the BBC reporting this... but after looking into it, I think a VERY embarrassed retraction from the BBC is not far away. It appears that Opeyemi Enoch has simply uploaded a paper onto academia.edu and then claimed the solution to the Nigerian media (which duly accepted it as "fact"). The paper Enoch uploaded has a different author's name (Raab) and was first published back in 2013. Manning (talk) 10:34, 17 November 2015 (UTC)
A bit more digging. The paper mentioned above was originally uploaded onto Bundlr in 2013 by Werner Raab (a retired mathematician from the University of Bonn). However on reading the paper, it appears to be a much older "proof" by an author who was apparently a Jesuit (the paper concludes with the letters AMDG or Ad Maiorem Dei Gloriam). The paper has no citations later than 1966. I suspect Raab simply uploaded it to the (non-academic) Bundlr as a curiosity, and not as a serious claim of proof. Manning (talk) 10:49, 17 November 2015 (UTC)
It's already been said that we don't report every time someone claims to have proven the Riemann hypothesis. The arxiv and its crackpot cousin vixra are littered with claims of proofs. Even the claimed proof by de Branges only gets a sentence or two, and even that was somewhat controversial to include (see the talk page archives), despite it being very well-known in the mathematics community, including commentary by Peter Sarnak. There is no way that a claimed proof would carry sufficient weight, without peer review and commentary by experts in the field. 12:12, 17 November 2015 (UTC)
The mention of de Branges in the article is there because it is RSed and meets DUE, and as a bonus passes RECENTISM. As such, the brief mention has not been the least bit controversial. The only "controversy" was one editor kept trying to put in OR to update the statement, bizarrely claiming a 2005 source supported a statement about a 2015 preprint. I kept removing it. The discussion has not been archived yet. See #Sarnak official Clay Math statement on RH above. Choor monster (talk) 17:06, 29 November 2015 (UTC)
Yes, I agree. You kept removing the reference to his 2004 claimed proof. (By the way, you can find that here).) 15:47, 30 November 2015 (UTC)

Be warned - this story has now been picked up by several other (generally reliable) sources, meaning that more people are going to try and add it to the article. Manning (talk) 14:22, 17 November 2015 (UTC)

A search for this in the news today shows that the story has spread to more media outlets, but debunking stories are also starting to pop up, and the BBC World Service story has been updated to indicate uncertainty. --2404:130:0:1000:6999:461F:AAAF:5C08 (talk) 06:04, 18 November 2015 (UTC)

## Semi-protected edit request on 17 November 2015

Solution to the Riemann Zeta hypothesis

A proof of the Riemann zeta hypothesis was presented by a Nigerian academic, Dr. Enoch Opeyemi on November 11, 2015 during the International Conference on Mathematics and Computer Science in Vienna, Austria becomes more symbolic coming on the exact day and month 156 years after the problem was delivered by a German Mathematician in 1859.

Dr Enoch first investigated and then established the claims of Riemann. He went on to Consider and to correct the misconceptions that were communicated by Mathematicians in the past generations, thus paving way for his solutions and proofs to be established. He also showed how other problems of this kind can be formulated and obtained the matrix that Hilbert and Poly predicted will give these undiscovered solutions. He revealed how these solutions are applicable in cryptography, quantum information science and in quantum computers. Ojengwa (talk) 10:51, 17 November 2015 (UTC)

Extraordinary claims require extraordinary sources. Given the number of purported solutions of the Riemann hypothesis including any claim of solution in the article will need to be backed by high quality sources indicating the solution is being taken seriously. If other mathematicians review this claim and respond favourably then we can look at this again. Hut 8.5 13:41, 17 November 2015 (UTC)
See comments above. This is yet another clear indication of the unreliability of news organizations on scientific matters. I think WP:RS should be adjusted to clarify this. 13:59, 17 November 2015 (UTC)

## Semi-protected edit request on 18 November 2015

The Riemann Hypothesis was solved in 2015 by the renowned Nigerian professor named Dr. Opeyemi Enoch [1]

Kenluck2001 (talk) 06:00, 18 November 2015 (UTC)

• Not done for now: The reference you have provided says that he has "claimed" to have solved it. That does not mean he has solved it and it has not been verified yet. If and when it has been verified the information can be put in. --Stabila711 (talk) 06:12, 18 November 2015 (UTC)
I like the way that the BBC source says "The interview was conducted with Dr Enoch on the basis that his solution is correct and that he has won the prize". The BBC should know that claims to have solved Riemann are ten a penny and that only acceptance by the academic community and winning the Clay prize would constitute a proof.--♦IanMacM♦ (talk to me) 06:22, 18 November 2015 (UTC)
• Some of the media coverage of this is weird. For example, this story in the Irish Independent is headlined "Professor becomes a millionaire after cracking 150-year-old maths conundrum" although there is no evidence that he has won the Clay prize or similar. Many mainstream media articles have failed to make clear that this is only a claimed proof at present.--♦IanMacM♦ (talk to me) 06:45, 18 November 2015 (UTC)
Yes. These sorts of stories make clear the unreliability of mainstream media for technical subjects. Which is why, as Slawomir said above, we need sources with mathematical expertise. —David Eppstein (talk) 06:54, 18 November 2015 (UTC)
FWIW, I've started a thread at WT:RS. It seems like we need a clear, bright-line, guideline about this issue. It seems like news media can no longer be regarded as reliable sources for scientific matters. Period. 16:48, 18 November 2015 (UTC)

## The "Enoch solution" - status

For the benefit of editors wishing to add the news of the claimed "solution" by Dr Opeyemi Enoch, here is why Wikipedia should not be in any hurry to alter the article.

1. - We have been here many times before. Because of the prize money and notoriety of the Millennium problems, people regularly claim to have "solved" them. There have been hundreds of claimed solutions since 2000 and only one of those turned out to be true.
2. - Even when the BBC says it, we are not beholden to publish claims that are demonstrably wrong. In this case (and regardless of whether his proof is correct) it is a fact that Dr Enoch has NOT been awarded the Millennium Prize. By their own rules, a solution must be published in an peer-reviewed academic journal and accepted by the wider mathematics community for at least two years before awarding a prize can even be considered. For the only prize awarded so far, seven years elapsed between Perelman's publication in 2003 and his being awarded the Millennium Prize in 2010.
3. - Dr Enoch's proof is apparently getting published in December 2015 and this is the first opportunity that the wider mathematics community will have to examine the proof. If they DO accept it, there will be PLENTY of media coverage of that fact - as was the case with Wiles/FLT and Perelman/PC. (Just for the record, FLT was not a Millennium problem, but it was a notorious problem with many claimed-but-wrong solutions over the years).

Manning (talk) 03:11, 19 November 2015 (UTC)

I strongly disagree with this view. Wikipedia is an encyclopedia for all, not just for the mathematical community. I am interested in mathematics but am not a mathematician. I had heard about the Riemann Hypothesis but am unfamiliar with the details so when I read about Dr. Enoch's solution, I was curious what it was all about and was surprised not to find even a mention of him in this article. I agree that "hundreds of claims" cannot be mentioned in this article, but very few of such claims reach mainstream media such as CNN or BBC. People reading BBC or CNN and interested to know more go to Wikipedia and will be very surprised not to find anything because the mathematical community within Wikipedia considers it unworthy of mention. It may very well be a false claim, but by adding a small paragraph to the article on Dr. Enoch's claim and explaining how his claim should be considered would make this page a lot more up-to-date and more accessible to the larger public. And that's what Wikipedia is all about. Loranchet (talk) 09:52, 19 November 2015 (UTC)
The fact that this claim has got coverage in mainstream media does not in itself make it more worthy of attention. It means that the claimant has managed to come to the attention of the media and that news organisations think it's a good story. Very few journalists actually know anything about the subject matter, so this doesn't make it any more likely that the solution is correct or is going to be accepted. Wikipedia writes its articles by relying on what reliable sources say, and news organisations are not considered very reliable sources for this type of material, particularly when they are not merely summarising something published in an academic journal. Hut 8.5 10:26, 19 November 2015 (UTC)

## Computations

Platt's work (2011 and later) should be added. Also, shouldn't we say something about the degree of rigour of some computations? I've heard that Gourdon-Demichel is a bit ropey on this point, due to non-rigorous sampling. Garald (talk) 22:43, 30 January 2016 (UTC)

As with many other WP articles on physics and math, there is quite a high "baseline level" of knowledge required for the reader. This is counterproductive, as an encyclopedia is supposedly intended for people who do not already know everything about the subject (or else why would they look it up?).

Example from this article: in the section "History", the function "Li" is introduced without any explanation or even link to an explanation. The term "Li" is short enough to have two meanings in math listed in WP: https://en.wikipedia.org/wiki/Polylogarithm and https://en.wikipedia.org/wiki/Logarithmic_integral_function. Thus, being a relative novice and not knowing which one is referred to (although I can guess), I can't even edit in a link here. Please have some perspective and put yourselves in the situation of someone less well educated, but still keen to learn. If you don't, I will suspect that you are more interested in showing off your expertise than in spreading knowledge. Sorry. Wdanbae (talk) 07:31, 2 February 2016 (UTC)

You didn't see the sentence "The function Li occurring in the first term is..." ? It's a few lines down, but all of the lines between the formula and that sentence are also explanations of other parts of the same formula. —David Eppstein (talk) 08:04, 2 February 2016 (UTC)
Sorry, my bad. Although I would have preferred to have the explanation even earlier than that in order not to lose faith, I guess I am oversensitized by all the other occurrences of unexplained or unlinked-to terminology that I have encountered elsewhere. Here, statement withdrawn. Wdanbae (talk) 08:57, 2 February 2016 (UTC)

Sorry for asking something that's not really related to Wikipedia, but here goes. I recently put on line a paper I wrote commenting on the attempt of Louis de Branges to prove the Riemann Hypothesis. How can I bring it to the attention of those who would be interested? Eric Kvaalen (talk) 09:10, 2 February 2016 (UTC)

## One of the references points at a non-existent page

Mazur, Barry; Stein, William (2014), Primes. What is Riemann's hypothesis - the link http://modular.math.washington.edu/rh/ brings up a 404 when I try to navigate to it. I did a quick look for it, but I wasn't 100% sure exactly which book it was supposed to be. Perhaps someone more knowledgeable than me can point the link towards a working location?142.109.6.1 (talk) 19:09, 11 February 2016 (UTC)

I've updated it to a working link for the same book under a slightly different title and date, http://wstein.org/rh/David Eppstein (talk) 19:51, 11 February 2016 (UTC)

## Popular expositions

Does it really make sense to break out "Popular expositions" as a separate subsection of the references section? Also, I am puzzled why some things were included and not others. The selection seems rather arbitrary in my opinion, and not really connected with any reliable metric of "popularness" of sources. I think WP:NPOV should urge against this arbitrary judgement of some sources to be "popular" (and thus, less good perhaps?) 15:18, 24 April 2016 (UTC)

I completely agree with this separate subsection. Maybe some other works should be included in it also (I haven't gone through all references yet), but these four definitely are - euphemistically - "popular" expositions. And they are (not perhaps but certainly) "less good". In fact I wonder if we should not suppress these references from the article. None of the authors is a specialist of the subject. Sabbagh is not even a mathematician, and his book is his one and only work reviewed by MathSciNet (by D.R. Heath-Brown, who politely reports that it is not a good book). The report on du Sautoy's book states that it is "not sufficiently accurate and complete". The report on Rockmore's book is not good either (also from Heath-Brown). Finally Derbyshire (who is credited with only 3 reviewed works on MathSciNet!) has the worst review ("While some chapters are not too bad, most seem to miss their mark"[…] "I am not sure the author ever answers the question of why the Riemann hypothesis is important.") Sapphorain (talk) 17:12, 24 April 2016 (UTC)
I don't think they should be suppressed, and I think calling them "less good" is an oversimplification, but I agree with keeping them separate. They have a different character than the other references and are aimed at a different audience. Keeping them separate helps that audience to find material on this subject that is readable by them, and helps mathematicians avoid material that is too popularized to be useful to them. —David Eppstein (talk) 18:04, 24 April 2016 (UTC)
I almost agree to conservation. Rather reluctantly concerning Derbyshire's book. But I don't agree at all concerning Sabbagh's books. I just moved in the new subsection his other book on the subject (also published in 2003…), which is not reviewed by MathSciNet and Zbl (and not even indexed by MathSciNet!). Here is journalist, who is writing vulgarization on about everything you can think of, who is not a specialist of the subject, who is not even a mathematician, even in a very large acception of the term, and who publishes two books on the Riemann hypothesis the same year. This is not serious. Sapphorain (talk) 22:39, 24 April 2016 (UTC)

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## Fast Prime?

Sorry if this is well know, but looking at the patterns in 3d (hard to view 4D) of the values of non-prime numbers, it seems like one could simply look at the data around the chosen point to follow it down to 1/2. I guess a proof is (a, all 'fingers' lead down to 1/2 eventually and we know that their are an infinate number of primes.... so proving infinate fingers may be a step. I do apologize if I'm an idiot. I'm no maths whiz but for 20 years this problem has held me in it's grip. My only wish is for SOMEONE to prove, or disprove OR fall into Gödel's incompleteness theorems.