Talk:Riemann hypothesis

January 2016: new year, same old story
Re this edit: Pioneer Scientific Publisher is yet another website that accepts pretty much anything, and people have to pay for the privilege of doing so ($400 to $850 according to the source).-- ♦Ian Ma c M♦  (talk to me) 09:29, 15 January 2016 (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)

Please make it readable
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 Here, statement withdrawn.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)
 * i mean the riemann hypothesis as a concept is already pretty complex of a topic (no pun intended) so that level of knowledge of math as stated in the original post should be expected for the most part, and for those who dont know that stuff yet are wanting to learn the riemann hypothesis then they should be expected to learn those stuff on the way. and though putting the note that Li is referrign to the logarithmic integral function could be put a little early its not all that big of a deal as the logarithmic integral function is already extremely closely tied to the riemann zeta function so its really easy for many people to assume its the logarithmic integral function and not the polylogarithmic one. 50.47.91.133 (talk) 17:32, 21 March 2022 (UTC)

Asking for suggestions
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)


 * Any chance you'd be willing to put up a PDF copy? The PNG method is quite inconvenient and slow.  If you're not willing to make one generally available, you can email me a copy through Wikipedia.  I can't promise I will make any cogent comments, but parts of it do look rather interesting.   Sławomir Biały  (talk) 13:41, 3 January 2017 (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?)  S ławomir Biały  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. — Preceding unsigned comment added by 213.106.56.145 (talk • contribs)

Failed attempts

 * 1) try to connect mathematically the trivial zeros to the non trivial ones  — Preceding unsigned comment added by 2A02:587:4114:9400:486E:F7D4:172C:1629 (talk) 21:11, 17 December 2016 (UTC)
 * 2A02:587:4114:9400:486E:F7D4:172C:1629 should speak more clearly. In the theory of analytic functions, all zeros are equally important. He has just volunteered for the task of connecting. — Preceding unsigned comment added by 212.159.119.123 (talk) 13:14, 3 January 2017 (UTC)
 * Is this a good place to mention https://arxiv.org/pdf/1602.03553.pdf from 2017, yet another attempt? I'm sure it isn't a sound proof or I would have heard of it via other channels, but I was hoping to somewhere find an explanation of what's wrong with this one. Arxiv has some reputation, but I think the short answer would be to file it somewhere within the eight attempted proofs currently included in the article. (I wish I were sufficiently mathematical to be able to do so, but...) Shanen (talk) 20:59, 8 March 2022 (UTC)
 * The arXiv moderators classified this one as "General Mathematics (math.GM)". That generally means they thought it was junk. The long version history is also a bit of a red flag. I have no specific knowledge of what might be wrong with it, though. In any case we'd need secondary sources by other people about it in order to add coverage of it here; there's a place in the world for cataloguing all the crank proofs of crank-magnet topics, but this Wikipedia article isn't it. —David Eppstein (talk) 21:14, 8 March 2022 (UTC)

An "In Popular Culture" section?
Anyone think a "In popular culture" section would be appropriate for this very serious article? Wolfram has three and I'm assuming there are more: http://mathworld.wolfram.com/RiemannHypothesis.html — Preceding unsigned comment added by BashBrannigan (talk • contribs) 15:45, 25 February 2017 (UTC)


 * I tend to agree with WP:TRIVIA that such sections are a waste of time and tend to fill up with uninteresting, unsourced, and insignificant anecdotes. —David Eppstein (talk) 17:02, 25 February 2017 (UTC)


 * WP:POPCULTURE sections aren't banned, but they can be enormous trivia magnets, and can end up like this cartoon parody showing how not to do it. When something like this is worth mentioning, it can usually be incorporated into the main body of the article.-- ♦Ian Ma c M♦  (talk to me) 17:29, 25 February 2017 (UTC)

Quantum system found
A quantum system has been found whose energy levels correspond to the zeros of the zeta function. This system was proposed on 30 of March 2017 in Physical Review Letters by Carl Bender of Washington University in St. Louis, Dorje Brody of Brunel University London and Markus Müller of the University of Western Ontario. See this general audience article:

https://www.wired.com/2017/04/maths-1000000-question-isnt-just-mathematicians-anymore/

2001:44B8:266:D05:85D0:EEF3:C950:5AC (talk) 13:25, 30 May 2017 (UTC)


 * I would not object to mentioning this in the article.  Sławomir Biały  (talk) 13:33, 30 May 2017 (UTC)

Proportion of zeros on the critical line
On 14 June 2017, N. Robles and K. Pratt have submitted a paper that claims to improve the lower bound for the proportion of zeros on the critical line to 41.49%. See https://arxiv.org/pdf/1706.04593.pdf, and the authors page at http://www.math.uiuc.edu/~nirobles/research.html. I hesitate to include the result in the Wiki article before it is peer reviewed, but it's worth to keep an eye on it. If their method turns out to be valid then there is further room for improvement: As they mention in footnote 2 on p.20, they were able to push the bound on the length $$\theta$$ of Feng's mollifier beyond $$\tfrac{17}{33}$$, to $$\theta<\tfrac{6}{11}$$, and they think that (with some more effort) a bound of $$\theta<\tfrac{4}{7}$$ could be reached. This in turn would further improve the lower bound on the number of zeros. Renerpho (talk) 13:46, 16 June 2017 (UTC)
 * Note that it is often said that Levinson produced a proof for 33.33% and Conrey for 40%. — Preceding unsigned comment added by 31.53.52.243 (talk) 07:21, 14 July 2017 (UTC)

Proof?
http://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170306.17.pdf — Preceding unsigned comment added by 134.191.220.76 (talk • contribs) 11:08, 13 August 2018 (UTC)


 * The International Journal of Theoretical and Applied Mathematics is another example of an online vanity press online facility where anything can be published with the appropriate payment.-- ♦Ian Ma c M♦  (talk to me) 12:36, 13 August 2018 (UTC)


 * The Science Publishing Group (which is the editor of this journal) is contained in Beall’s list of predatory journals and publishers. Sapphorain (talk) 13:38, 13 August 2018 (UTC)

Michael Atiyah
Re this edit: Michael Atiyah has claimed a proof and there is a good deal of discussion about what it is. Since there is some WP:CRYSTAL here, wait and see is the best policy.-- ♦Ian Ma c M♦  (talk to me) 02:23, 22 September 2018 (UTC)
 * As far as I can tell, anyone who might have an informed opinion about this is saying "no comment" until details become available. I agree, there is no need to rush to include this. —David Eppstein (talk) 03:41, 22 September 2018 (UTC)
 * Having watched the talk (and followup questions) live with several people mildly knowledge in the area, I do not strongly believe that Atiyah has successfully proved it, and agree with David Eppstein that we should wait for reliable sources. Enterprisey (talk!) 09:20, 24 September 2018 (UTC)
 * The video of Atiyah's lecture is on YouTube here and lasts 49 minutes. The audience and the press seemed less than impressed .-- ♦Ian Ma c M♦  (talk to me) 17:18, 24 September 2018 (UTC)
 * He talks about the "proof" starting about 35:55. Bubba73 You talkin' to me? 00:21, 25 September 2018 (UTC)
 * Something carefully written should be added as the number of viewers jumped from a few thousands to 50K. — Preceding unsigned comment added by 193.224.79.1 (talk) 06:19, 25 September 2018 (UTC)
 * We don't write articles based on viewership trends. There are an awful lot of attempted proofs of the Riemann Hypothesis which have not been accepted, so far this isn't distinguishable from them.  Hut 8.5  06:45, 25 September 2018 (UTC)
 * & How about now? :) - Astrophobe (talk) 01:37, 10 February 2020 (UTC)
 * Huh? (BTW I saw your edit on my watchlist, but changing a comment to add a ping after-the-fact doesn't work.) By now it's clearer that there was no proof. We already mention this, very briefly and with appropriate skepticism, in our biography of Atiyah. I don't think it rises to the level of needing to be mentioned here. —David Eppstein (talk) 02:07, 10 February 2020 (UTC)
 * Wow, I would never have thought that it would distinguish between adding a ping to an existing edit and making a new edit with a ping. OK. - Astrophobe (talk) 02:17, 10 February 2020 (UTC)
 * The interest of other mathematicians in Atiyah's claimed proof seemed to fizzle out quickly. It is quite a long time ago now and the prize offered by the Millennium Prize Problems is intact.-- ♦Ian Ma c M♦  (talk to me) 07:24, 10 February 2020 (UTC)

Solution of Riemann hypothesis
I solved them. I want to submit them Bipul Kumar Jaiswal (talk) 07:28, 20 December 2018 (UTC)


 * Wikipedia is not the place to do this, as it cannot accept original research. It's also worth noting that there are many proofs on various predatory open-access publishing websites, and these are not accepted by the academic community.-- ♦Ian Ma c M♦  (talk to me) 07:34, 20 December 2018 (UTC)

Twin prime conjecture
I found location of twin prime number. I will want need some help A belong to integers then, A is not equal to BC+B+C then, A is infinite Please say this statement is right/wrong. If statement is right please send it prove. If your proof is right then, I join your name for solution for twin prime conjecture Bipul Kumar Jaiswal (talk) 08:13, 16 February 2019 (UTC)


 * As pointed out previously, Wikipedia cannot help with original research. The conjecture that there are infinitely many twin primes is also beyond the scope of this article.-- ♦Ian Ma c M♦  (talk to me) 11:21, 16 February 2019 (UTC)

Explanation of s
My suggestion is to explain or to reference what 's' stands for in the Riemann zeta function. — Preceding unsigned comment added by 2A02:A455:91CE:1:6570:9FA9:6C2:B880 (talk) 11:36, 13 November 2019 (UTC)
 * What part of "whose argument $$s$$ may be any complex number other than 1" do you think fails to be a proper explanation? Or did you mean, why does our article use that particular letter for the argument, rather than $$x$$ or $$q$$ or whatever? If so, you do realize that the choice of letter is arbitrary and does not affect the mathematics, right? —David Eppstein (talk) 18:32, 13 November 2019 (UTC)
 * What part of question did you not understand? It seems clear to me that the question revolved around the historical reasons for choosing s. While it goes without saying (which makes me wonder why you felt the need to point it out) that the choice of variable (not "letter") does not effect the mathematics, it may not have been arbitrary.  Do you know that it was arbitrary for sure?  What is your source and why didn't you provide it? 208.93.202.97 (talk) 16:59, 9 September 2022 (UTC)
 * s is a sort of standard choice for complex-valued variables, just like x for real-valued and n for natural- or integer-valued variables. I think this usage came from its use here with the zeta function. - CRGreathouse (t | c) 20:22, 9 September 2022 (UTC)
 * Riemann's original article is linked as an illustration at On the Number of Primes Less Than a Given Magnitude. It is evident that he used s, and that he did not provide an explanation for using s. Wovon man nicht sprechen kann, darüber muss man schweigen. —David Eppstein (talk) 21:00, 9 September 2022 (UTC)

None has, or none have?
Re this edit: we could debate this all day long, despite it having nothing to do with the Riemann hypothesis. According to this source "OED says: Some traditionalists maintain that none can only take a singular verb (as in none of them is coming tonight rather than none of them are coming tonight). However, none is descended from Old English nan meaning ‘not one’, and has been used for around a thousand years with either a singular or a plural verb, depending on the context and the emphasis needed." The answer seems to depend on the source, and there is no definitive right or wrong answer.-- ♦Ian Ma c M♦  (talk to me) 06:17, 15 April 2020 (UTC)


 * Thanks IanMacM; in my experience with mathematical English, "none are" is much more common than "none is" (in agreement with David Eppstein's revert). --JBL (talk) 13:44, 15 April 2020 (UTC)

Pathetically bad article
This article contains a great deal of advanced material, in almost all cases very poorly explained, and it doesn't even have a list of some of the smaller zeroes of zeta(s) on the critical line.

Quality is more important than quantity.

Just in case anyone cares that readers understand what is written here, this article needs a great deal of work.50.205.142.50 (talk) 17:03, 20 June 2020 (UTC)
 * Your claim that it doesn't list the smaller zeros is factually incorrect. The three smallest are in the caption of the lead image of the article, and more are in the "Gram points" section. —David Eppstein (talk) 18:53, 20 June 2020 (UTC)


 * Mathematics is a very incremental subject. Most advanced topics are difficult or impossible to understand without first understanding a pile of precursor topics which a typical person won't be familiar with. In the case of the Riemann hypothesis a reader who has no knowledge of complex analysis, infinite series or number theory isn't going to get very far, and that includes almost everybody who hasn't studied mathematics at university level. We could write an article along the lines of the ones in Category:Introductory articles which tries to explain the topic and underlying concepts to people who have no mathematical background, but it would have to be about as long as this article and it would need to be a separate page.  Hut 8.5  18:59, 20 June 2020 (UTC)

Reverted edit about Tag systems
Regarding this edit, by User:David Eppstein, I would like to disagree that the statement is entirely vacuous, and ask for suggestions to improve it. Yes, I agree that since Tag systems are computationally universal, it holds that there would theoretically exist a system which could prove or disprove the Riemann Hypothesis. However, the importance of the statement, is that such a system has been constructed. Similarly, on the Busy beaver page, there is a section on notable examples which states that a Turing Machine has been constructed which halts iff the RH is false. These statements are not vacuous even though it follow from the Church-Turing thesis that they must be true. The actual construction of the Turing machines mentioned in the article are entirely within citations in the article. I similarly left a citation to an article which shows the actual construction of the tag system I was referencing. Could you please suggest ways I could improve my edit, such as formatting the construction of the tag system in a Wiki-friendly manner, or perhaps this information belongs on a different page, instead of striking down valuable additions? Floridada (talk) 16:10, 12 March 2021 (UTC)
 * Floridada appears to exist only to add WP:REFSPAM to Wolfram to articles. And the fact that a computationally universal system can verify a counterexample to Riemann is not something that needs a tag system to construct. A tag system is merely an obfuscated way of doing it. Wikipedia is not an obfuscated code contest. —David Eppstein (talk) 16:58, 12 March 2021 (UTC)
 * I'm sorry that you don't appreciate the importance (historically, and mathematically) of interesting systems such as Tag systems, and have channeled your disinterest into attacking the intention of my edits instead of reviewing the content I add. Sure, I happen to cite a lot of Wolfram material, because I find their content very interesting. I suggest you give the article I initially cited a read, it might convince you of the importance of these less-appreciated systems. In the meantime, could another editor comment on the possible addition of a statement regarding Tag systems in this page, and whether Eppstein is editing from a non-Neutral Point of View regarding his clearly strong bias against Wolfram material, which is hurting the neutrality of articles like this by keeping out interesting, factual, well-supported edits? Floridada (talk) 18:07, 12 March 2021 (UTC)
 * The edit was not an improvement, for the reasons DE mentioned in his original edit summary. --JBL (talk) 18:57, 12 March 2021 (UTC)
 * Perhaps I can elaborate why I think this is irrelevant here, though. I think the universality of tag systems, proved by Minsky in 1961, is worth knowing, but after that point the main use of tag systems is to prove other things universal or undecidable. When you reduce Riemann to tag systems, you're going the wrong direction, taking a problem that might be decidable somehow (Riemann) and reducing it to a problem that we know is undecidable (termination of even certain simple concrete tag systems). It doesn't help us resolve Riemann because tag systems in general aren't an easy way to solve things, and it doesn't prove hardness of anything because for that you need to translate in the other direction from the known-hard to the unknown. So there is no hope that this kind of translation can be useful as a general solving-problem approach. —David Eppstein (talk) 19:26, 12 March 2021 (UTC)


 * The sourcing at https://www.stephenwolfram.com/ isn't ideal.-- ♦Ian Ma c M♦  (talk to me) 19:10, 12 March 2021 (UTC)


 * I have to agree with David Eppstein, the statement added is pretty trivial and if the best source available is a passing mention in a blog then it's not significant enough to include here anyway.  Hut 8.5  19:13, 12 March 2021 (UTC)

For those following along at home, Floridada has now been blocked, as one member in a farm of Wolfram-promoting sock-puppets. All the socks have been active for years :(. --JBL (talk) 19:22, 12 March 2021 (UTC)

"log" to "ln"
I think "ln" is a better notation than "log" for the following reasons:
 * Consistency: At present this article uses both log and ln.
 * Compliance with standards: ln is unambiguous contrary to log, probably more widely used, and recommended by ISO.

The only argument in favor of "log" seems to be that "log" is more accepted in the field of number theory. I think this argument weighs less than the previous two.2A01:CB00:A34:1000:ED0E:D7C1:D5BD:64BC (talk) 13:57, 13 June 2021 (UTC)
 * It may be the only argument but it's a strong argument. Log is much more standard in this area. "ln" is hard to distinguish from "1n" or "In", especially in text-based math formatting which we sometimes use. As for ISO, pfeh. Nobody follows ISO (see e.g. Binary logarithm for a more egregious example of ISO logarithm-notation that is best avoided). Consistency can be achieved by getting rid of stray instances of ln from the article. —David Eppstein (talk) 17:18, 13 June 2021 (UTC)


 * It's a number theory article, of course it should follow number theory conventions and use log rather than ln. - CRGreathouse (t | c) 15:38, 23 June 2021 (UTC)
 * ln is log_e, lg is log_10 and log is log_2. That is common and how it should be. Valery Zapolodov (talk) 18:09, 20 July 2021 (UTC)


 * Most scientific calculators use ln for logs in base $e$ and log for base 10, so I'm not sure if this is correct.-- ♦Ian Ma c M♦  (talk to me) 18:32, 20 July 2021 (UTC)
 * Also lots of computer scientists use lg for log_2. In any case this is a mathematics article and mathematicians use log for log_e; that's the convention we should be following. —David Eppstein (talk) 18:54, 20 July 2021 (UTC)
 * Did you just use imgur? Really? Anyway, log is implying that you should write the base, WHICH ONE cannot do on a calculator (still using it in 2021??). Full stop. See https://en.wikipedia.org/wiki/Logarithm "scientific calculators", BTW, my calculator also has log for log_10 but it also has normal log_x^y that I always use. "log for log_e" No. This is not a scientifical article. For example, WHAT is this? https://en.wikipedia.org/w/index.php?title=Riemann_zeta_function&diff=1034012285&oldid=1034010066 before it was obvious it is log_e. Now, it is not, AT ALL. "As for ISO, pfeh. Nobody follows ISO." Russia, France, et cetera follows ISO. It does not matter anyway, since wikipedia MUST follow ISO. lg is 10 base, log is 2 base. Full stop. "follow number theory conventions" "mathematicians use log for log_e", so what is correct? I think you need to add "USA" there too. Anyway, Wikipedia is neither. Valery Zapolodov (talk) 03:17, 21 July 2021 (UTC)
 * Whatever gives you the idea that Wikipedia must follow ISO? I don't know of any policy, guideline, or other precedent that would say so. —David Eppstein (talk) 04:51, 21 July 2021 (UTC)
 * I'd never given this much thought until now, but it is clear that there is no generally accepted way of specifying the base of a logarithm, and that a certain amount of personal preference becomes involved. Pretty much all scientific calculators use the scheme in the Imgur image.-- ♦Ian Ma c M♦  (talk to me) 06:57, 21 July 2021 (UTC)
 * Why should this be relevant? How much use do you think research mathematicians make of scientific calculators? —David Eppstein (talk) 07:01, 21 July 2021 (UTC)
 * Probably not much, but it was given as an example. The key issue is whether to use ISO notation, which specifies ln for logs in base $e$. When written in sans-serif typeface, it might not be as easy to read. The word Ill can also suffer from this problem. The previous word is ill with a capital I, but it is very hard to distinguish from the Roman numeral III (3) when typed like this.-- ♦Ian Ma c M♦  (talk to me) 08:00, 21 July 2021 (UTC)
 * I just came here searching for this topic. Anyway, since ln is a bit confusing and there's not consensus on the meaning of log, shouldn't loge be preferable for clarity? DavoDovox (talk) 12:06, 20 August 2023 (UTC)
 * See arguments above about following standard notation rather than making up notation that nobody uses. —David Eppstein (talk) 12:13, 20 August 2023 (UTC)

New claimed proof from India
Re this edit: it is sourced here   (among others in the Indian news media today) but this brings to mind Opeyemi Enoch in 2015.-- ♦Ian Ma c  M♦  (talk to me) 17:54, 28 June 2021 (UTC)


 * This needs vastly better sourcing before it's included here. Both those articles claim that the solution to the Riemann hypothesis "could open the doors for the use of primes in cryptography" (as if that wasn't done in the 1970s), and say the proof has been "verified" by some physicists.  Hut 8.5  18:02, 28 June 2021 (UTC)


 * Perhaps this source should work. It's listed as reliable at WP:RSP. &#8212;&#x202F; Vaibhav afro &#x202F;&#128172; 02:52, 29 June 2021 (UTC)


 * I'm worried about WP:NOTNEWS and WP:10YT here. Opeyemi Enoch's claimed 2015 proof fizzled out and hasn't been heard of for a long time. Kumar Eswaran has managed to pick up media coverage for his claimed proof, but this is not the same as convincing other mathematicians and winning the Clay Mathematics Institute prize.-- ♦Ian Ma c M♦  (talk to me) 06:54, 29 June 2021 (UTC)


 * Mainstream media outlets, even higher quality ones, aren't great sources for mathematical content, see Reliable sources. This particular article seems to be a rewrite of a press release outlining the discovery, so it doesn't add anything over the initial press release. Which incidentally comes from a teaching institution which (judging from their website) doesn't appear to do any research. Exceptional claims require exceptional sources. A proof of the Riemann hypothesis would be the greatest mathematical discovery in some time and because of that plenty of people have falsely claimed to have proved it. A real proof would generate far more coverage including from better sources.  Hut 8.5  07:49, 29 June 2021 (UTC)


 * This article from The Quint provides some necessary details, Dr. Easwaran's area of expertise is far away from these fields. TrangaBellam (talk) 15:52, 29 June 2021 (UTC)
 * So, this is the proof and this is the 'roadmap'.
 * His brother is quite miffed and writes, Nuff said.
 * DEBUNKED. TrangaBellam (talk) 16:08, 29 June 2021 (UTC)
 * To save everyone from needing to wade through this: it seems that this is actually one of the better claimed proofs of RH. Although both sloppy in its notation and ultimately wrong—they're all wrong—when cleaned up, it does yield a valid heuristic that RH should be true -- a known heuristic, alas, from 1968. - CRGreathouse (t | c) 13:26, 13 August 2021 (UTC)
 * A heuristic which is also very well known to the point where a variant of it is already in this article. JoshuaZ (talk) 12:48, 13 August 2021 (UTC)

Short description
@Ianmacm Please read the short description instruction page if you haven't done so. Two key quotations (emphasis in the original): The current shortdesc that you have reverted to violates point #1 because it attempts to be a definition. The fact that "mathematical conjecture" might be a suitable shortdesc for many articles does not make it unsuitable for this article. Best, Wham2001 (talk) 21:09, 27 March 2022 (UTC)
 * 1) Each short description should: be short – no more than about 40 characters (but this can be slightly exceeded if necessary)
 * 2) A short description is not a definition and should not attempt to define the article's subject nor to summarise the lead.
 * No. You are making a confusion. The current shortdesc doesn’t attempt in any way to define the article. It just attempts to provide something more than a totally useless information on it - which is what your version provides. It is of no help whatsoever to describe as a « mathematical conjecture » what is indeed clearly a conjecture concerning a subject that is indeed clearly mathematics. A better way to be useful would be to provide a concise and elegant formulation conveying the same information - no more, but no less. --Sapphorain (talk) 21:37, 27 March 2022 (UTC)
 * In particular, two of the main functions of short descriptions are (1) to help disambiguate search results, on mobile, and (2) as part of annotated lists of links in see-also sections. In both cases (for instance, when searching for things named after Riemann or in mathematics articles where one might see this in a see-also section) the fact that this is mathematical is a given, and the fact that it's a conjecture is so vague as to be meaningless and unhelpful. Putting them both together adds pleonasm to the mix: conjectures are always mathematical. So User:Wham2001's proposed shortdesc "Mathematical conjecture" is bad. It's also based on a misreading of WP:SDNOTDEF: SDNOTDEF does not mean "say something extremely vague in an attempt to avoid saying anything about the topic of the article", it means "aim for understandability and conciseness in explaining what the article is about rather than aiming for a pedantic precise definition". What one needs is something indicating that this is about the locations of the zeros of the zeta function. The status quo, "Conjecture in mathematics linked to the distribution of prime numbers" is more informative, but still doesn't really pinpoint what this is about (the connection to prime numbers is pretty indirect), and is also too long (69 characters when the target length is 40). The zeta song gets it better in the first line: "Where are the zeros of zeta of s?" But maybe that's too informal. How about "Conjecture on zeros of the zeta function", exactly 40 characters? —David Eppstein (talk) 23:21, 27 March 2022 (UTC)
 * I stand by the edit summary saying that "mathematical conjecture" is too vague. It needs a bit more detail and I will go along with what others think is suitable.-- ♦Ian Ma c M♦  (talk to me) 06:48, 28 March 2022 (UTC)
 * I agree that "Conjecture on zeros of the zeta function" is much better than my attempt. Thanks! Wham2001 (talk) 06:51, 28 March 2022 (UTC)
 * "Conjecture on zeros of the zeta function" is excellent. TrangaBellam (talk) 06:55, 28 March 2022 (UTC)

Prime gaps
Looking at the relationship between RH and prime gaps: It is true that Crámer proved around 1920 that the size of a prime gap is $$O(p_n)$$ (assuming RH). But it is also true that he went on to show around 1930 that the size of a prime gap is $$o(\log^3 p_n)$$, as part of a stronger but more specialized result. See Granville's "Harold Crámer and the distribution of prime numbers" for a discussion of the history. It seems strange to me that the article mentions the former but not the latter. Perhaps I am missing something? Russ Woodroofe (talk) 16:33, 13 April 2022 (UTC)


 * Are you talking about result just above (11) in that paper? It allows o(x/log^4 x) exceptions, so the largest prime gaps could be larger. In fact even (11) itself can't show sqrt-sized gaps: set y = sqrt(x) and note that there could be on the order of log^2 x gaps of this size. - CRGreathouse (t | c) 05:20, 14 April 2022 (UTC)


 * Right! Missed the exception, and in my head I'm often very sloppy up to asymptotic density 0 exceptions.  (I am certainly not a real number theorist.)  I guess that it would still possibly be due to include in the article something about consequences of the RH for typical prime gaps, but it is not immediately obvious to me where to fit it in.  Now, the thing that made me bring this up is the treatment in the article of this result of Dudek.  From what I can tell, this is a mild improvement of earlier work by Ramaré and Saouter (which itself builds on earlier work).  It looks a bit WP:UNDUE to me to list the result with the current level of context. Russ Woodroofe (talk) 08:12, 14 April 2022 (UTC)

Plot of Re and Im part
Good morning @David Eppstein, Sorry, yes you're right: the plot cuts off at 30. Can we provide a better plot? The current one seems misleading, showing zeros along the red line at 1, 2x near 14, 2x near 21, 2x near 25, and 1x just under 30. Thanks, Hansmuller (talk) 06:34, 1 October 2022 (UTC)
 * Those are not zeros. They have imaginary part zero (the blue curve) but nonzero real part (the red curve), or vice versa. The zeros are where both colors cross the axis at the same point. —David Eppstein (talk) 06:35, 1 October 2022 (UTC)
 * Thanks,of course that was silly of me. Perhaps we can indicate this explicitly, to help other mistaken fools like me.Thanks again, Hansmuller (talk) 06:39, 1 October 2022 (UTC)
 * @David Eppstein So i was wrong again. The upshot of the graph wil now be mysterious to many.


 * Could you please briefly indicate in the caption where we can find the zeros in the plot, if it is not at crossings for zero values? Thank you, Hansmuller (talk) 06:48, 1 October 2022 (UTC)

Limits to provability?
This article seems to consider only two possibilities: either that the RH is true and we can easily prove it to be so, or that it is false and we can easily prove that.

However, as Kurt Gödel showed, sometimes things aren’t so cut-and-dried. There are undecidable problems in mathematics; one of them, in fact, is the problem of deciding whether a given computable real number is actually zero or not.

So, does that mean the RH could actually end up being somewhat ambiguous? (I’m thinking, what if it turns out that 1. there are values away from the critical line that _could_ be zeroes, 2. it is possible to show that no algorithm can prove that any of these values is _actually_ zero rather than just being a near miss, and 3., conversely, it is also possible to show that there is no way to prove that _all_ of the “candidate zeroes” are near misses). If 1., 2., and 3., are all true, this would make the RH permanently ambiguous.

Has this possibility been seriously considered (or, alternatively, ruled out?) The article should probably mention any “grey area” the Riemann hypothesis has, if in fact there actually is any.

An “ambiguous” Riemann scenario would have to result in all of the corollaries being similarly ambiguous (for example, with something like Robin’s inequality, there would have to be instances where the inequality is possibly violated, but no algorithm can prove if it actually is, or if it merely “bumps up against” the threshold for Riemann violation without crossing it. While at the same time there’d be no way to prove that _all_ such instances are just near misses). 2600:1014:B07A:D001:F13B:C6F7:2888:1A08 (talk) 02:43, 15 August 2023 (UTC)


 * This issue is surveyed by Matiyasevich "The Riemann hypothesis from a logician's point of view" (see also "The Riemann Hypothesis in computer science",  who credits Turing 1939 for first studying it and Kreisel 1958 for proving that RH is equivalent to statement that some decidable property holds for all $$n$$, the first level $$\Pi^0_1$$ of the arithmetical hierarchy. (The part of our article that states "A related bound was given by Jeffrey Lagarias in 2002" is also of this form.) That means that if RH is false, it is provably false, but does not imply provability in the case that it is true. —David Eppstein (talk) 06:40, 15 August 2023 (UTC)
 * OK, thanks. 2600:1014:B07A:D001:F13B:C6F7:2888:1A08 (talk) 13:06, 15 August 2023 (UTC)

Junk
Someone added a bunch of junk to this page, which could be AI-generated but is obviously inappropriate regardless. John Baez (talk) 21:54, 8 March 2024 (UTC)


 * Looks like User:Ttwaring has cleaned up the mess. —David Eppstein (talk) 23:59, 8 March 2024 (UTC)

Conjecture is better than hypothesis
On the third paragraph it would be better to use "conjecture" instead of "hypothesis". This is a conjecture and when proved it will become a theorem.

Instead of "The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that..." I would write "The Riemann conjecture is concerned with the locations of these nontrivial zeros, and states that..." Zeyn1 (talk) 17:48, 24 April 2024 (UTC)


 * The Riemann hypothesis is the WP:COMMONNAME so there is no need to use an alternative wording.-- ♦Ian Ma c M♦  (talk to me) 18:00, 24 April 2024 (UTC)


 * I think its called the Riemann hypothesis because that's how it's used: as an extra hypothesis to prove results that would otherwise be unattainable. Regardless, it's the common name so we're not changing it. - CRGreathouse (t | c) 12:29, 26 April 2024 (UTC)

Picture could be improved
The article includes a picture of zeta(1/2 + it) for -N ≤ t ≤ N.

This is an important illustration for this article.

Unfortunately, it was not made with a 1:1 aspect ratio.

For this purpose, there is absolutely no reason to depict the complex plane with anything but 1:1 aspect ratio.

I hope that this picture can be replaced by one with the appropriate aspect ratio oof 1:1.