Talk:Riemann hypothesis
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Nash
Can someone comment Nash's words about zeroes of Euler - Riemann zeta function ζ(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)
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: [1] (http://zdnet.com.com/2100-1104_2-5229702.html?tag=zdfd.newsfeed) - Someone claims to have proven the Riemann Hypothesis. I picked this up off of /. (http://science.slashdot.org/science/04/06/09/223241.shtml?tid=134&tid=98&tid=99) - 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 theoream is extremely important in prime number theory. →Raul654 02:49, Jun 11, 2004 (UTC)
- There was a lengthy explaination 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)
External link
I have cut out this one ("WhatPC?" Article (http://www.whatpc.co.uk/news/1157891) 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"[2] (http://www.math.purdue.edu/ftp_pub/branges/apology.pdf)(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" [3] (http://www.math.purdue.edu/ftp_pub/branges/riemannzeta.pdf)(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. [4] (http://arxiv.org/abs/math.NT/9812166) 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 (http://www.wbabin.net/aladeh/riemann.pdf) --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: [5] (http://reed.claremontmckenna.edu/Colloquium/laurel-beckett04082003.asp)[6] (http://www.uic.edu/depts/ims/classschedule/SPRING2005/HON.htm)[7] (http://www.cs.nyu.edu/pipermail/fom/1999-July/003254.html)[8] (http://sunsite.utk.edu/math_archives/.http/hypermail/historia/may00/0122.html)[9] (http://www.ms.uky.edu/~jrge/340/)[10] (http://economics.wustl.edu/~bparks/agre/agre.pointers.html)[11] (http://mathforum.org/library/drmath/view/51929.html) - Zondor 22:12, 13 May 2005 (UTC)
Another proposed proof
I'm moving this off the page to here instead. Does anyone know if this has been seen or reviewed by the rest of the mathematical community?
- On 18 March 2005, Fayez Fok Al Adeh, the President of the syrian Cosmological Society, published on the Site of "The general Science Journal" an astonishing simple proof of the Riemann Hypothesis.(A Link to the proof) (http://www.wbabin.net/aladeh/riemann.pdf). But a proof of such an hypothesis must be very critical verified.
- I read/skimmed it. Its ugly and nasty and broken. Its no proof at all. Its 10 pages long. Pages 1-7 (equations 1-49) are very simple, freshman-calculus manipulations of simple integrals. Rather tediously dull. Lets assume these manipulations are correct (although I spotted an error/typo on page 2.) But then equation 50 starts getting strange and equation 56 is just plain bizarre. After 7 pages of freshman calculus, there's a sudden (mis-?)application of a variational principle of some kind, without any ado or explanation, as if it were just some more basic calculus. Equations 57, 58 and so on don't follow, don't make sense, and nothig after that is any good, although the manipulations continue in the same freshman-calc type of presentation. Its junk. linas 00:21, 15 Jun 2005 (UTC)
And what about DeBrange's proposed proof, has it been considered debunked or warranting more look into it or what? - Taxman Talk</sup> 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 (http://www.maths.ex.ac.uk/~mwatkins/zeta/RHproofs.htm) 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.
Xavier Gourdon and Patrick Demichel
Re the recent revert; Gourdon gives thanks to Demichel in his paper, but the wording makes it sound as if Demichel is a computer sysadmin/technician. linas 22:34, 14 Jun 2005 (UTC)