Talk:Riemann hypothesis/Archive 2

Nash
Can someone comment Nash's words about zeroes of Euler - Riemann zeta function &zeta;(s) that its zeroes are singularities of space and time. This one is of course from Howard's film Beautiful Mind. XJam [2002.03.24] 0 Sunday (0) [[ Nash Encounter with Georg Friedrich Bernhard Reimann (answer to XJam) courtsey: "A Beautiful Mind", Sylvia Nasar... Refer(chapter 32-secrets)

Nash was considering the pseudoprime sequences for each one of these one can associate with 'zeta function', which for for the case of the true prime reduces to Riemann zeta functions. Nash claimed to be able to show for 'almost all' of these pseudoprime sequences the corresponding zeta function satisfies the Riemann Hypothesis. [Venkat.s.iyer]

Analytic continuation
Sorry to be so dumb, but I never understood how -2, -4 etc can be zeroes of the function. Why isn't zeta(-2) just the sum of squares 1+4+9+... ? [2002.08.28] Stuart Presnell


 * Because obviously that would not work ! zeta(s) is defined using (sum 1/z^s) over all complex numbers s=x+iy with x > 1, then extended to the whole complex plane (excepted at -1) using analytic continuation. That is, zeta is the unique analytic function (= holomorphic function) on the complex plane (less -1) that matches the sum where it is defined. That is described in the first paragraph of the article on the Riemann zeta function. See also "analytic continuation". -- FvdP Sep 5 & 7, 2002


 * Ah, thanks! I was relying on the little knowledge I had gleaned from a few popularisations - didn't know about the 'analytic continuation' aspect of it.  If only I had bothered to read the appropriate Wikipedia entry...  [2002.09.13] Stuart

Divergence and meaning
Please forgive my ignorance, since I am not really into mathematics (in fact only 13 years old), but don't all inputs less than or equal to 1 in the zeta function cause it to be equal to infinity? Ilyanep 14:37, 9 Jun 2004 (UTC)

No - the infinite series is actually meaningless in that region; there are some other ways of representing the function, which allow one to discuss it there.

Charles Matthews 15:30, 9 Jun 2004 (UTC)

De Branges (I)
An interesting looking article: - Someone claims to have proven the Riemann Hypothesis. I picked this up off of /. - Xgkkp 23:25, 9 Jun 2004 (UTC)

De Branges - many people's hearts might sink. Of course the Bieberbach conjecture business counts in his favour; but still. Charles Matthews 07:20, 10 Jun 2004 (UTC)

Can anyone explain why anyone cares?
We state this is an important unsolved problem, but I have no idea either before or after reading the article why it's important. What makes this problem important? Yes, I know it will probably take about 3 pages to say why, and that it will have to be grossly oversimplified, but it would be nice to have some idea of what it would mean if [1] it were proved true or [2] it were proved false. - Nunh-huh 02:34, 11 Jun 2004 (UTC)
 * What, for practical applications? Not any, yet. But, prime numbers are *extremely* important in cryptography, and this theorem is extremely important in prime number theory. &rarr;Raul654 02:49, Jun 11, 2004 (UTC)
 * There was a lengthy explanation in a book called The Music of the Primes (dunno if that's an article), which says that if the Riemann Hypothesis is false, then that means that the prime numbers have a certain order to them. It would mean that the 'prime number coin' is biased and doesn't have a probability of 'landing' on a prime 1/log(n) (to base e) times, that there is an order. And if there is an order to the primes, it kind of jeopardizes some encryption methods (such as RSA, a method based on the difficulties of factoring numbers) Ilyanep 02:56, 11 Jun 2004 (UTC)


 * Ah. For a non-mathematician, this is not "intrinsically obvious". I hope you'll add the explanation to the article. - Nunh-huh 03:01, 11 Jun 2004 (UTC)

I think crypto is not directly relevant - though it is constantly brought into discussions of number theory. It is more like this: what we can know in prime number theory depends on the extent to which there can be a 'conspiracy' amongst prime numbers (it would have to be very large prime numbers, another reason why crypto misses the point) which defeats the kind of reasoning that says they are entities largely independent of each other. RH is probably the deepest single, simple statement saying 'no conspiracy'.

Charles Matthews 05:46, 11 Jun 2004 (UTC)

About Applications - the prime number theorem is true; what is at issue is whether the error term is random walk-like (square root of the main term, or nearly) or bigger.

Charles Matthews 16:37, 11 Jun 2004 (UTC)
 * Okay, let me go fix that Ilyanep 16:42, 11 Jun 2004 (UTC)

Thanks for the explanations. Is there any chance that someone with more of the right mathematical background than I could translate it into nice-ish english? It would really help the article to have the significance of the hypothesis in there, in words that lay readers could understand :-) - Paul 21:15, 28 September 2005 (UTC)

LDH 00:14, 6 August 2006 (UTC) I like CM's answer in terms of "no conspiracy". Take a look at the statement about the Farey sequences, known to be equivalent to the Riemann Hypothesis. It's a pretty basic statement about the relationship between addition and multiplication of ordinary integers. And this is only one of many plausible and natural-looking statements that would follow from RH. If somebody proves RH, we will really know something.

External link
I have cut out this one ("WhatPC?" Article Article on how proof of the Reimann hypothesis could destroy E-Commerce). I think it has no useful content.

Charles Matthews 19:29, 7 Sep 2004 (UTC)

Agreed. No counter examples have ever been found even after much searching. It's proof would just validate what everyone thinks is probably true about primes. It's disproof might jeopardize cryptography. pstudier 21:56, 7 Sep 2004 (UTC)

And it might not have anything much to do with crypto, in fact. Charles Matthews 19:15, 20 Oct 2004 (UTC)

500 or 1000?
just a minor point, from the book "music of the primes" the quote from hilbert is waking up after "five hundred years" not a thousand. Searching on google seems to give conflicting answers though, anyone read the actual article ?.

Wombat 02:04, Dec 8, 2004 (UTC)

--This quoted from The Riemann Hypothesis, by Karl Sabbagh, Chapter 4, page 69, my edition "For Hilbert, the Riemann Hypothesis became the most important of all his problems, if we are to believe a story often told in mathematical circles: According to German legend, after the death of Barbarossa, the Emperor Frederick I, during a Crusade he was buried in a faraway grave. It was rumored that he was not dead but asleep, and would wake one day to save Germany from disaster, even after five hundred years. Hilbert was once asked, "If you were to revive, like Barbarossa, after five hundred years, what would you do?" He replied, "I would ask, 'Has somebody proved the Riemann Hypothesis?'"" And this citation: Bela Bollobas, foreword to Littlewood's Miscellany, Cambridge University Press, 1986, p. 16. Hope you can use this information - rob chamberlin 22:48, 19 Feb 2005 (UTC)

De Branges rumour
I'm moving this off the page. Nothing recent or newsworthy, I think. Charles Matthews 12:13, 12 Feb 2005 (UTC)

A possible proof of the Riemann hypothesis
In June 2004, Louis De Branges de Bourcia of Purdue University, the same mathematician who solved the Bieberbach conjecture, claimed to have proved the Riemann hypothesis in an "Apology for the proof of the Riemann Hypothesis"(pdf). His proof will soon be subjected to review by other mathematicians. De Branges de Bourcia has announced a proof a number of times, but all of his previous attempts at this proof have failed.

The full purported proof is "Riemann Zeta functions" (pdf).

The proof's method has been tried before unsuccessfully. Linked is Conrey and Li's counterexample on the problems in the earlier version of his proof.  The example involves a numerical calculation. The authors also give a non-numerical counterexample, due to Peter Sarnak. On the other hand, De Branges's successful proof of the Bieberbach conjecture was also preceded by his failed proofs of it.

i actually have no clue about riemann hypo. but this might be (and may be not) usefull for you Math gurus -> How to Prove the Riemann Hypothesis --82.102.204.79 19:33, 25 Apr 2005 (UTC)

Practical Uses of the Riemann hypothesis
The practical uses of the Riemann hypothesis include many equations that have been 'solved' in abstract mathematics with the assumption of the Riemann hypothesis.

Also, if there is a disproof of the Riemann hypothesis, it implies that the primes have a certain order to them. It would show if the error in the Prime number theorem is Random walk-like or not.

I removed this, which seemed vague and not well written; I'll stick a version of the first sentence as an opener to the consequences section. Gene Ward Smith 22:27, 14 Feb 2005 (UTC)

Riemann zeta hypothesis?
Who if anyone calls RH that? Charles Matthews 16:14, 13 May 2005 (UTC)
 * typo/informal/ignorance. here: - Zondor 22:12, 13 May 2005 (UTC)

Another proposed proof
And what about DeBrange's proposed proof, has it been considered debunked or warranting more look into it or what? - Taxman Talk 19:17, Jun 10, 2005 (UTC)


 * DeBrange is a real mathematician and his proofs are long and difficult. I think errors were found, but he's probably still workig on it.


 * If you want proofs, theres a bunch of them, Matthew Watkins has a collection of them at proposed proofs page Many of these are actually interesting to read; Castro & Mahecha for example. And no such collection is complete without a proof from the illustrius Ludwig Plutonium who provides not one but two proofs! linas 00:21, 15 Jun 2005 (UTC)

No one want to claim that DeBourge isn't a real mathematician .. but the length of a proof don't say anything about the proof.

De Branges's proof
I am restoring the mention of his proof. I haven't heard of it much recently so I am guessing the proof was somehow flawed. I have no idea about his reputation, but I think it's nice to elaborate on this for those who wonder if he has proven the hypothesis or not. -- Taku 03:52, July 18, 2005 (UTC)

http://mathworld.wolfram.com/RiemannHypothesis.html has a good summary of attempted proofs. Someone who knows this stuff might reflect that on this article. -- Taku 03:57, July 18, 2005 (UTC)


 * I don't know if there is a good policy on 'rumours'. My own view is that we can mention them in articles when they are new, and interesting. It seems proper to me to remove them after a time; or replace them with something neutral (for example, if there is a proof submitted to a top journal for refereeing, just say that). After a while, with no new developments, a rumour becomes little embarrassing to have in a major article. For example, it becomes better to write about De Branges and RH in the De Branges article. Charles Matthews 07:25, 18 July 2005 (UTC)
 * Isn't there any solid info on the status of the general acceptance of the proof? Are they really that arcane that it can take over a year to even decide if they are solid or not? If there have been some problems pointed out in the proof, we should remove it's mention until/if they are fixed. - Taxman Talk 18:57, July 18, 2005 (UTC)


 * Wiles' proof of Fermat's theorem took nearly a decade before it was widely accepted. linas

It depends on the importance of a hypothesis or a conjecture whether we should make some elaboration on the status of proofs. For obscure conjectures, I don't think readers are interested in proofs much. But, one like RH, I think there is a room for not just mathematical aspects but some social ones; e.g., A Beautiful Mind has a scene in which Nash is proving RH. Likewise, how people have tried to solve this should be interesting. We do agree that each article in wikipedia is for the general public, right? Of course, it remains open whether De Branges's noteworthy. At least I agree that his attack for RH has to be mentioned in his article. (Sorry for my lateness in reply) -- Taku 06:04, July 25, 2005 (UTC)


 * I don't think it should be mentioned in the article, when it has dropped out of newsworthiness. There have been past examples (Matsumoto, Vernon Armitage ...) and it is too much to cover them all. Charles Matthews 07:58, 27 September 2005 (UTC)


 * There is something more than newsworthiness to justify the presence of de Branges' proof in this article. The fact that no one has so far debunked it, and that de Branges is a renowned mathematician, seem enough to warrant him a mention. The past examples, I assume, are not worth mentioning precisely because they have been shown to be flawed. When the same becomes true for de Branges' paper, then Matthew Watkins will add it to his list, the paragraphs will be removed and de Branges' connection to the subject will be safely reduced to a cursory note. If, however, the proof is correct…


 * I think I have found a way to settle this question. I have replaced the paragraphs on de Brange's proof by a more concise version that still does him justice in that no one has so far disproved his arguments, and have meanwhile moved the original paragraphs to his biographical article. Incidentally, it is I who has made the preceding comment on newsworthiness and, for that matter, mine are both the previous version and the current one.


 * I reverted your replacement of the paragraphs by a more concise version because I hadn't seen your reasons. I found them now, so I've undone my revert. Sorry about that. Could you please use an edit summary in the future, so that we know the reasons for your edits? A short note like "see talk" would have sufficed in this case. Thanks. -- Jitse Niesen (talk) 06:55, 17 June 2006 (UTC)


 * I have summarized this section even further. It seemed reasonable, for I have realised that Mr. Matthews may have some some reason after all. It is unfair to give de Branges' unduly attention if there are other standing proofs like Castro's. The subject will have to await new developments until the article can suffer new changes. Personally, I believe Castro's work may be a spin-off of de Branges', for the latter's previous papers have been used by the former, as can be seen in the former's references. But this is pure speculation.

I have removed the link to the "Hilbert-Polya" proof for several reasons. Firstly, it is Castro and Mahecha's and can be found from Matthew Watkins' page. Secondly, I think it has been left implicit that no unverified proof should be highlighted. Thirdly, the link provided leads to an error page. Castro and Mahecha have posted a new, corrected version of it in arXiv, and its address has changed. I can understand the point of giving some emphasis to their proof, as it follows a historical proposition, but this could be made on the corresponding page. Come to think of it, I shall add a note in Hilbert-Pólya conjecture.

To Prove or Not to Prove
Perhaps I am misunderstanding something. If the Reimann Hypothesis is true, how would that suggest that there is no order to the primes? All the non-trivial zeros would lie on the critical line. That's order! What if the Reimann Hypothesis is true and someone could still find order to the primes? I don't understand this. I thought if someone found a proof for the Reimann Hypothesis, then that would mean he or she has found the order of the primes. Could someone explain?

213.190.42.200 10:15, 23 July 2005 (UTC)


 * I'll take a stab at this. The primes are sort of unpredictable, but there are equations governing their distribution.  If the Riemann hypothesis is true then there are tighter bounds in equations about their distribution. Bubba73 01:54, July 27, 2005 (UTC)

---

The proof of the Riemann Hypothesis means to get the best possible estimate of the quantity of prime numbers under a given value. It doesn't mean, that the prime numbers have an order, but they are lying in an appreciable area. Then the zeroes on the critical line have the real part 1/2, but the imaginary part has still no order as the prime numbers have no order in the hitherto sense. W.A.

0.5?
Yesterday I replaced three or four instances of "1/2" by the HTML code &frac12;, but I wonder if 0.5 would be better? Bubba73 00:33, August 2, 2005 (UTC)


 * I don't mind the "1/2" and I guess the &frac12; is OK, but please no 0.5. linas 21:33, 2 August 2005 (UTC)

Clarify joke?
This paragraph is obviously a joke, but this really should be clarified. If it's not a well-known joke, it should be removed altogether:


 * Number theorists as a rule have lived a long and productive life, thus it is said that the person who proves the Riemann hypothesis will actually become immortal. Others say that Riemann's hypothesis has been proved false several times already, and every time the unhappy mathematician was struck dead as soon as he had finished the proof.

Thanks. Deco 04:51, 22 August 2005 (UTC)

Xavier Gourdon
This article currently gives top billing to Gourdon, making mention in the intro paragraphs, while completely ignoring dozens(?) of equivalent formulations. (e.g. the class conjectures from diophantine equations, or the Li criterion) Why? Surely the achievements of the numeric calculations are not so great as to merit this kind of prominence, is it? linas 04:21, 23 August 2005 (UTC)

Carlos Piedrahita
There should be an annoucement of the review of DeBranges paper as soon as possible. What isa happening ?

Disproof of Riemann's hypothesis?
Chun-Xuan, Jiang of the China Institute for Basic Research has published a disproof of Riemann's Hypothesis. Does someone know, if this paper is checked? 14:40, 15 November 2005 (UTC) Kaufmann Friedrich

--- I don't know, but this "disproof" looks very strange because no analytical extension is explicit used (I couldn't find one when I looked through this paper cursorily). They assume the extension, that's all. I don't understand the sentence, that the computation of all zeroes are in error, because it's proofed that there are zeroes on the critical strip. In this disproof I miss the domain for the variables. 15 November 2005, Winfried Aschauer


 * No zeroes at all in the critical strip, eh? Charles Matthews 16:07, 15 November 2005 (UTC)


 * Who knew? Who could have guessed? I found the introduction quite remarkable, given what follows. Anyway, equation seven is wrong, and its hard to pay attention after that. Also, defining m_-(t) and m_+(t) as the "number of primes which..." is wrong, since one or both of these will be infinite. linas 17:24, 15 November 2005 (UTC)


 * Indeed, the introduction is something... Anyway; I've personally discovered a trivial proof, but I'm keeping it to myself. Fredrik | tc 17:38, 15 November 2005 (UTC)


 * I have a proof that fits in the margin of this WP page. However, try as I might, I seem unable to actually write in the margins! linas 23:26, 15 November 2005 (UTC)


 * Oh please write down your proof and show it to the world, it's NOT possible that it is worse than the bosh produced by Jiang Chun-Xuan ! W.Aschauer


 * Haven't they proved that an infinity of zeros are in the critical strip? Ozone 05:38, 11 January 2006 (UTC)


 * Yes; all are in the criical strip (except the "trivial" ones). There's a proof that 1/4 or 1/3 of all zeros in the strip are actually on the line. Don't know the details. linas 02:12, 12 January 2006 (UTC)

---

It's somehow funny, that they don't like to have zeroes on the critical line. I have spent about 10 minutes to read this text. Every second was wasted time. They don't care about domains. They use the Euler-Product as they like to have it. Defined or not defined - it does no matter for them. What a rubbish. 19:58, 15 November 2005, Winfried Aschauer

You need to also take into considerantion the venue: this is Hardonic Press, the mouthpiece of a famous nutcase, Maria Santilli Ruggieri who "dicovered" Hardonic Mechanics and who has "proven" Einstein wrong. Multiple times.


 * I know: What shouldn't be cannot be.