# Talk:Hilbert–Pólya conjecture

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Field:  Number theory

## Fredholm stuff

I tried to cleanup the edits by User:Karl-H but failed. In particular, this needs an equation for the kernel.linas 22:26, 5 May 2006 (UTC)

## Concrete attempts

I have added a cursory note on Castro's and Mahecha's proof. I have done it for two reasons. Firstly, it settles the affair that has taken place in the RH page on how much emphasis should be put on unverified proofs (among which Castro & Mahecha's and Louis de Branges' are the most prominent); I deem it adequate to level it down to a wide mention. Secondly, it does justice to this nhistorically justified approach. Similarly, I have removed all specific mention to de Branges' proof and added it to his biography.

The fact is that it is a preprint, and written in quite unconvincing language ... Charles Matthews 15:36, 18 September 2006 (UTC)
Even not being a mathematician myself, I quite agree with you about the language. The way they open the article seems most suspicious to me. It differs somewhat from de Branges, whose purported proof uses no rhetoric and is as laconic as any article in my own area. However, the paper is their third version (if I understood well the nomenclature of arXiv), the first one being from 2002, so they seem pretty much confident of their results. What is more curious is that they include de Branges as one of their consulted references, and this has led me to ask whether their proof is independent. Their original paper does predate de Branges' announcement by more than a year, but the tools de Branges uses belonged to the public sphere at least since Sabbagh's book on RH was published (and in fact long before that; de Branges developed his theory of Hilbert spaces of entire functions back in the sixties, and his approach to RH dates back to the late eighties). De Branges' own proof is, of course, surely independent, as it stems from his own previous work. About the language, may I add that Perelman's papers, which I have not looked at myself, are described as having been written in a most unconventional way, some parts of it by hand. The content is what matters, isn't it? In a side note, I agree with you that all these proofs are to be taken with a grain of salt. I think the two I mentioned are the most important because their authors cannot be described exactly as cranks. De Branges obtains his proof from the field he developped with this purpose, and Castro and Mahecha are physicists that declare themselves to be versed in functional analysis and differential operators. But I do agree that we ought not to create an article or make a fuss about every "open" proof posted at Matthew Watkins' page. But to dismiss them for good I would like someone would come up and say, like in the case of Arenstorf's proof of the twin prime conjecture: "Lemma X in de Branges' proof is incorrect, and so is Lemma Y in Castro & Mahecha's". The fact no mathematician has taken the trouble to do it so far is quite galling to me. I concede that mathematicians must know better if they choose to ignore the papers, but are we really having to wait for their proponents to recant them or die? (De Branges is not exactly young)
Physicists aren't 'cranks', but who goes to a physicist for a proof? There have been very many attempts on RH, and in a sense if there was any good chance this was correct, it would have had more attention and criticism by now. Charles Matthews 14:51, 19 September 2006 (UTC)
You are quite right about they not being cranks, but the Hilbert-Polya conjecture is described as being intimately related to quantum mechanics, so I would expect physicists to be familiar with this particular branch of mathematics. Anyway, I have always thought that in the most theoretical areas it became progressively more difficult to tell a physicist from a mathematician. But I agree with you about attention from the mathematical community. Whereas in the case of de Branges the people able to read his proof are counted in the fingers of a hand (I have done so me research and found out that VERY few people are conversant with the theory of Hilbert Spaces of entire functions), and this would justify two years of estrangement, Castro & Mahecha used, from what I could find out by inquiring, more basic tools of functional analysis that most mathematicians would be able to scrutinize. Personally, I believe that their proof is probably flawed, but it would still be good if someone came and said something like "their differential operator is inadequate because of this, this and that". Of course, you could still be right that the simple fact that they are physicists justifies the scepticism, but shouldn't we offer them the benefit of the doubt?
• I agree with the user before, we always must give the benefit of the doubt when you present a proof, for example all the mansucripts presenting a proof of RH should be verified saying lemma or theorem in line xx is false , we should ask ourselves what would have happened if referees mathematicians or physicist had been so rigorous or intolerant with Newton Einstein Heisenberg and others (when they were unknown and young) then perhaps we would still be living in stone age, they blame (mathematicians) "Spanish Inquisition" and Church but they act the same way.

The main problem is the prize,.. this won't make things easier to check since there will be lots of "greedy" people inventing proofs or using misconceptions just to say they have proved RH to become famous , so any serious attemted proof will be dismissed and turned down without taking any look at it, appart form De branges Castro and Macheca i have seen proofs involving Hamiltonians H=T+V where the author gave the potential that still remain ignored, published in Mathew Watkins' page ,but ignored by the scientific community --Karl-H 19:09, 26 September 2006 (UTC)

Please sign your posts. No, the 'benefit of the doubt' is for criminal law cases, not analytic number theory. Charles Matthews 18:53, 26 September 2006 (UTC)
Dear Mr. Matthews, if the right not to sign one's posts is enshrined by Wikipedia, I believe it is up to each one's discretion to decide on it. Personally, I like anonimity, even if this implies that the IP address I am writing from is recorded. Of course, I leave it up to you to decide whether you want to engage in discussion with a sphinx, but, in the Internet, what is really a name? For all you know, I could be de Branges, signing as Perelman, writing from your computer when you are not looking.
As for my comment on the benefit of doubt, I think you are being a little restrictive here. I have already conceded that mathematicians are right to regard purported proofs with suspicion. My comment was specifically that this should not apply to the authorship of the proof. It was you who brought up the fact that Castro and Mahecha are physicists, and this seemed to me the worst possible reason to dismiss a proof. After all, Fermat was a lawyer. I am not going to repeat what I have written above on the subject of credibility and the likelihood of their proofs being correct. I do believe that mathematicians must know better if they choose to ignore them, but that does not mean that all mathematicians will behave like this forever. In other words, the estrangement these papers have received cannot avert the fact that they are still subject to doubt, and doubt, philosophically, always favours the proponent, so it is not as if you or any mathematician had a choice. So long as nobody comes to say that their proofs are incorrect because of this or that (of course, I am not talking about cranky proofs, but am specifically referring to papers that are so intrincate that no one can dismiss them with a short look), they may be correct for all we know. I find it a little mean that my comment has been taken out of context. I expected better. Actually, your hasty and laconic answer could be easily taken for - forgive me - supercilious patronizing.
As for the prize, may I point out that de Branges has developed his theory of Hilbert spaces of entire functions in the course of half a century with the expressed purpose of solving the Riemann Hypothesis and similar questions (he was announcing his previous attempts long before the Clay Mathematics Institute was founded). He is a bit old and too renowned to be transported by the possiblity of a prize (after all, one million dollars is not that much for someone who holds tenure, owns an apartment in Paris, and is approaching the end of his life, if you come to think of it; it does not even suffice for him to restore the Chateau de Bourcia, as someone has written in his Wikipedia biography). Castro and Mahecha is another story, as people have been wondering. Castro has a record of controversy (some of his papers have been refused even by arXiv), and their proof - which cites de Branges' previous works - was published after the prizes were announced without any previous research background on the subject, which, as Mr. Matthews seems to believe, does indicate that their proof may be a canard. I also believe de Branges, as Atle Selberg has warned about obsessive people, may have been obfuscated by his long-sought goals and lost his objectivity in many accounts. Still, he is not causing any upheaval, continues to work quietly and has sought no publicity, which speaks on his behalf (I do not count Sabbagh's book as publicity; he approached de Branges, not the contrary). In the research proposal published on his site, he even goes so far as to volunteer to verify Castro & Mahecha's proof, something everyone has refused to do so far (and considering that their paper predates his, something that may even put Castro and Mahecha in the limelight as the first proponents instead of him, if their proof turns out to be correct). I may be a naive outsider, but this seems quite reasonable and much sounder than the current estrangement. And may I remind you that even Matthew Watkins, who has compiled all proposed proofs, cranky and professional alike, states that it would be a shame if number theory students did not waste a few days of work to deconstruct them so they do not go unverified, and goes so far as to suggest that instructors set this as a task.
Finally, you seem to take for granted that attention and criticism are warranted for any proposed proof that has a chance of being correct. I have addressed this in my previous post.
I'm not quite sure what your point is. Signing here with ~~~~ is conventional good manners, and gives a time stamp so that one can follow the development of the thread. As for proofs of RH, there is a normal process to follow in having a proof accepted by a learned journal. As the case of FLT shows, this is a perfectly good route even for a breakthrough result. I wouldn't recommend anyone the course of trying to get publicity, otherwise. There have been numerous claims made and retracted (Armitage, Matsumoto, ...). Charles Matthews 14:02, 27 September 2006 (UTC)
Double dots are just as good a form of keeping a thread. And you need not the four tildes, as you can check my IP at the history page of this conversation. And no timing is needed, as one posting must necessarily follow the other in the course of time (it would be quite ludicrous if I added an answer before your comment, instead of after). Good manners, I believe, are much more in the content than in the form. As for the conventional route of peer-review, yes, I do agree with you, and I have often wondered myself why de Branges and others have not followed this route. The only answer I can offer is that mathematics is not the firmest footing for peer-review. Verification takes a long time in this area, and some mathematicians (including people like Perelman) have resorted, for their own inscrutable reasons, to other forms of publication. In the case of FLT, as you certainly know, Wiles was wise enough to gather momentum, to create an interest around the subject he had developed in secrecy before submitting the article. Virtually nobody knew what he was going to talk about when he scheduled the seminars in Cambridge. Only after the standing ovation did he supply the proofs of his article for verification. You may well reach the same conclusion some people I have talked to have reached: if you submit an abstruse proof of a famous conjecture using techniques only you and a handful of others master, any journal editor will be as hard put to find readers as yourself are, and refusal will ensue. Arenstorf tried the same with his failed proof of the prime number theory, and it seems to be a trend in mathematics to go fishing for interest in the preprinting realm, in order to make sure that, on submission, the paper will not be neglected. I do not know if this is the case of the purported proofs of RH, but I think it may be at least in the case of de Branges, for the reasons I have discussed above. if I still need to make myself clearer, I am not defending the contents of the proofs. I cannot analyse them for myself. What I am discussing here are possible patterns of behaviour from the people involved and from mathematicians at large. And as for retracted claims, well, they have been retracted. Neither Castro and Mahecha nor de Branges have followed this course of action after a couple of years, so their proofs are still under the umbrella of doubt.
May I just add that so far we have more agreed than not. It was on your initiative that I have curtailed the mention to purported proofs of the Riemman Hypothesis from its page and added them cursorily to ancillary ones, like the present article and dee Branges' biography. We are not even talking about the subject that initiated this discussion. We are actually discussing a very unsubstantial topic, and I do not think you are opposed to the comment on Castro and Mahecha's proof in this article (otherwise, you would have reverted it to its original form). As for what my point is, let me rephrase it tersely: "There are two purported and unverified proofs of RH. They are probably flawed, but cannot be dismissed with a short glance. They have not been retracted in the course of four years. They are unconventional in publication, but this has become commonplace in the mathematical community. Their proponents have problems of credibility. Even so, they deserve cursory notes in corresponding articles of Wikipeadia. Full stop". Reasons and points of view may be found in previous posts. You either agree or do not.

Well, I may understand Wiles rather better than most people (we started research in the same office, not that we were ever very close). Please take into account the standard header I have added to this page. Our policy is strictly limited to documenting what people claim (at best). Convention seems to be that we may add 'rumours' when they are fresh; but will remove them later if there is no consensus. This is any case directly opposes the idea that long periods in limbo add to interest. Charles Matthews 09:36, 28 September 2006 (UTC)

We are not really talking about Wiles' motivations, are we? I have merely pointed out that nonstandard announcements of proofs are quite commonplace in the field. And, by the way, de Branges' proof (or the others, for that matter), is no rumour. A rumour would be: "Someone has told me in the lobby of the number theory department that a mathematician somewhere in Louisiana has found a proof of an important conjecture concerning prime numbers". The proof is announced, and has not been verified, so it is a very tangible thing. Actually, its incorrectness is merely a rumour: "Sarnak has declared that the proof is probably cobblers"; "Conrey believes the approach is incorrect", and so forth, without anyone stepping forward and saying "I have checked it and it is incorrect in its assumption that the obstacle imposed by the Euler product can be averted; my arguments are this, this and that". (Of course, I am not a mathematician, so this is just gibberish to serve as an example.) I do not understand what you mean by newsworthiness: this is an encyclopaedia, not a news service. I agree that the limbo does not add to interest, but you are incorrect in assuming that it is enough to dismiss a paper. Mendel's work on genetics rested im limbo for 35 years. The limbo, as it were, is precisely that: a standstill, a deadlock, something that can only be dissipated when someone sheds light over it. Their being ignored by the mathematical community may not add to the interest, but it does not detract completely from the proofs. It is strange that you and so many others seem to believe that de Branges and Castro & Mahecha have committed some sort of crime in avoiding peer-reviewed publication in the earliest stages. They are not even complaining publicly about their estrangement, and seem to be aware that only formal publication will bring credence to their work, if it is correct after all.
Anyway, I have only added a neutral comment, which I believe must stay there until it may be safely removed (or expanded in case of correctness). After all, nobody complained when a purported refutation of the Hodge conjecture, which had been published in the non-peer-reviewed arXiv, was mentioned in the corresponding Wikipedia article. Incidentally, it was I who removed it when the authors recanted the announcement, and I am going to do the same if de Branges or Castro follow that course of action. It seems we are having two different conversations here. As you have said, there is no breach of policy (how I hate that word) in mentioning what people claim, if the people are conversant with the subject.
• The fact that a paper has been published into a "peer-review" journal doesn't tell you it's correct, i would like to remind you the hoaxes of Alan Sokal, Peter Lynds and Bible codes that were published in worldwide known journals, in the end referees publish what they like , In the end all people have prejuices and interests so the evaluation of this or that paper isn't fair or objective , at least in Arxiv.org whereas you have an endorser (if you belong to a known university they assign you one) you can publish your paper on the condition they are correct in math or are verifiable, the general policy of a journal should be this if this is verifiable or mathematically correct or consistent then it worths publishing ,otherwise i consider neglecting a good paper a kind of censorship. (Please sign as requested)
I don't think this leads anywhere. While Mendel's work was neglected, we could not necessarily have changed the position. We are not supposed to lean towards 'outsiders', and also not supposed to show prejudice against them, either ; NPOV is about neutrality. The business about Sokal etc. is irrelevant. WP is not run on the same lines as academia. The lines on which is it run are called 'policy', because it has to be called something. People who talk about 'censorship' here are almost always quite wrong. We have to use a certain amount of source criticism to compile articles, as common sense would indicate. Charles Matthews 18:36, 2 October 2006 (UTC)

Once and for all: they are not always worthy of notes here. There can be no presumption that claims are encyclopedic. And I am not required to address anybody's arguments, point by point. Charles Matthews 18:45, 3 October 2006 (UTC)

Well, stalemate. It was you who first conceded (and actually mentioned) that claims are the limits of coverage for an article in Wikipedia (do I need to quote?). I do not know what you mean by "always", for you have not established a reasonable criterion to tell when it is the case (unless it was hidden behind a lot of moonshine), and so my understanding is as valid as yours. I have not said, in any moment, that I am going to include any loony who claims to have solved RH. I agree with you one hundred percent that cranks like Ludwig Plutonium (is it really his name?) or failed attempts that have been dismissed or recanted (like Armitage's) ought to be kept away from corresponding articles. The two cases we have discussed are precisely those that seem - not only for me - least incredible (or most credible, if you are open to the possibility, but this is beyond the point). If Perelman's claim had gone unverified, knowing his solid background in the area of topology, I would undoubtedly have added a note on his work, and I can just picture you making the same objections. You would then raise your points of presumption and common sense (it is quite galling to be in the receiving end of such vague criticisms; it is a bit hasty to label something as presumption, as is to appeal to common sense, seeing that every atrocity has been committed in the name of common sense). Perhaps we ought to open an article on "Cranks, failed attempts, and unverified claims of RH", but franly, this would be a waste of time. There is nothing more to be discussed, and you may believe what you please. And I may remind you that in a discussion arguments are to be countered, as I have attempted to do with yours. If I say something and my interlocutor changes the point, I am left at a loss to know whether he has been convinced or not. Required you may not be, but it is - to quote you - good manners, and even more so because it transcends policy. But, as I with signing, you are indeed enshrined in your right to ignore my points. I admit that this discussion has been fruitful in that it has helped to keep the cursory notes where they are, but now we are finished we may relieve the discussion page and remove this thread altogether. Unless, of course, you still think we may found common ground to compromise. I am not engaging in edit warring, so I leave to you the decision to remove those notes or not, even after what we have discussed (or have we?).

It is not the custom to remove talk page discussion, for good reason. You say: it has helped to keep the cursory notes where they are. I don't think so. You are actually just being annoying by not signing. Charles Matthews 19:43, 8 October 2006 (UTC)

I see that you (or someone else) have removed the note. I could see it coming. My comment on the helpfulness of the discussion referred to the fact that you seemed to await the conclusion before deciding to remove it or not, somethig I thought quite reasonable. I am not reverting it, as I promised, but I deem it sheer stubborness, since you (or the person responsible for removing the note) were unable to defend your stance or to counter my arguments successfully. As for signature, I could not care less about how you feel. I have had polite, constructive discussions with other people (to the extent of recognizing when I was wrong) here without any of them complaining about the fact. And unsigned discussions between other people are quite easy to find. "When everyone seems to have a problem, maybe the problem lies with you." And I must say you are quite attached to custom. But by all means refrain from enlightening me on the "good reason" (oh dear, how many vacant expressions!), though: I have nothing more to say on this subject, and shall avoid this page for the near future. Well, it is quite tedious to talk with headstrong people, so let me say my valediction. A good-day to you.

The removal was by User:R.e.b., probably prompted by comments made by User:Linas and me, on Wikipedia talk:WikiProject Mathematics. Charles Matthews 16:43, 9 October 2006 (UTC)

But my question to Mathematicians would be this HOw could we be sure that something has solved RH by finding a HIlbert-Polya operator? , let's say that tomorrow or next week we see on a respectable (which doesn't always mean verifiable or correct) a differential or integral operator of HIlbert-Polya type proposed by profesor Mr. XXX then how could we be 100 % sure he has proved RH by finding this operator?, i mean we could calculate the first 1000000000000000000000000000000000000000000000000000000000 eigenvalues of his operator and check ${\displaystyle 1/2+iE_{n}}$ but how would we know that for example the 10000000000000000000000000000000000000000000000000000000124567890-th eigenvalue is different from what we would expect or that exist a possible eigenvalue so ${\displaystyle |\zeta (1/2+iE_{k})|>0}$ then the Hypothesis by Hilbert and Polya would be useless to prove or disprove RH

There is a recent publication (2017) that makes some progress towards this conjecture : DOI (10.1103/PhysRevLett.118.130201): Carl M. Bender, Dorje C. Brody, and Markus P. Müller, Hamiltonian for the Zeros of the Riemann Zeta Function, Phys. Rev. Lett. 118, 130201 – Published 30 March 2017. I do not have the necessary background to judge whether it should be added to this article. — Preceding unsigned comment added by 207.140.59.194 (talk) 19:39, 31 March 2017 (UTC)

## Check recent edits!

User:85.85.104.70 (who, by a WP:COI trail, appears to identify himself as 'Jose Javier Garcia Moreta') has made recent edits to this article. Many of his other edits have been unproductive, but I don't know this subject well enough to see if that is the case here. Would someone please verify these?

CRGreathouse (t | c) 22:59, 14 July 2010 (UTC)

## Where does Hilbert figure in this?

It may be a minor point, but aside from the name of the conjecture, Hilbert isn't mentioned in this article. Whether he didn't originate the idea, or if he was thought to have done so historically and our understanding has changed, shouldn't his role or lack thereof be mentioned? Stevko (talk) 09:42, 31 January 2012 (UTC)

## random matrix

In the relation to the random matrix theory the distribution of non-trivial zeros of Riemann zeta function is a GUE as described in this article. Meanwhile, GUE means that the time-reversal symmetry breaking. What is Meaning of the time reversal symmetry breaking in the number-theoretic viewpoint?--Enyokoyama (talk) 16:39, 21 April 2013 (UTC)

I add an reference by A. Connes and slightly correct the subsection "Recent times." The Connes work is a little earlier than the work of Berry and Keating written in the lower subsection "Possible connection with quantum mechanics."

If the zeros of ζ(s) can be the spectrum of an operator R=1/2・I-iH, where H is self-adjoint then H might have an interpretation of an certain physical system and a proof of Riemann Hypothesis might have been encoded in physics such as quantum mechanics. I think this is the origin of Hilbert-Polya conjecture.--Enyokoyama (talk) 05:50, 28 April 2013 (UTC)

Not only the proof of analogous formulas for Selberg zeta function by A. Selberg bu also the proof of Weil conjecure by Deligne may be viewed as a realization of Hilbert-Polya conjecture.--Enyokoyama (talk) 10:18, 11 May 2013 (UTC)