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Claimed Proof of Riemann Hypothesis 345

Posted by CmdrTaco
from the two-scoops-of-math-please dept.
An anonymous reader writes "Xian-Jin Li claims to have proven the Riemann hypothesis in this preprint on the arXiv." We've mentioned recent advances in the search for a proof but if true, I'm told this is important stuff. Me, I use math to write dirty words on my calculator.
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Claimed Proof of Riemann Hypothesis

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  • Dirty Words (Score:5, Funny)

    by Rik Sweeney (471717) on Wednesday July 02, 2008 @11:38AM (#24031621) Homepage

    Me, I use math to write dirty words on my calculator.

    Such as 80085?

  • by i_want_you_to_throw_ (559379) on Wednesday July 02, 2008 @11:39AM (#24031645) Homepage Journal
    By using Fourier analysis on number fields, we prove in this paper E. Bombieri's refinement of A. Weil's positivity condition, which implies the Riemann hypothesis for the Riemann zeta function in the spirit of A. Connes' approach to the Riemann hypothesis. Weather permitting of course. (Just looking on the positivity side)
  • by multipartmixed (163409) on Wednesday July 02, 2008 @11:39AM (#24031657) Homepage

    Man, where's Charles Eppes when you need something explained to you in layman's terms?

    • by Notquitecajun (1073646) on Wednesday July 02, 2008 @12:00PM (#24032073)
      Ummm...I think that WAS layman's terms. For you math geeks, try being a history major and looking at all that. It just looks like a cat walked on the keyboard to me...
    • Re:Tried to RTFA (Score:5, Informative)

      by PlatyPaul (690601) on Wednesday July 02, 2008 @12:17PM (#24032369) Homepage Journal
      The Riemann zeta function is \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} [written for LaTeX], or "the sum of 1/(n^s) as n goes from 0 to infinity (increasing by 1 repeatedly)" [in more human-readable form].

      Riemann was interested in the zeros to this function, where s is a complex number. He conjectured that all zeros (aside from those of the form s = -2c, where c is a positive integer) would have to be of the form (1/2) + ki, where k is a constant and i is the square root of -1.

      This paper is saying that they've found a way to verify this intuition by patching a hole in a previous attempt.

      Assuming that everything is correct (a big assumption), this would finally solve a long-standing problem (dating back to 1859).


      Details of the actual solution are a bit heavy. Those actually interested in this sort of number theory might want to start here [amazon.com].
      • typo (Score:5, Informative)

        by Ungrounded Lightning (62228) on Wednesday July 02, 2008 @12:24PM (#24032481) Journal

        The Riemann zeta function is \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} [written for LaTeX], or "the sum of 1/(n^s) as n goes from 0 to infinity (increasing by 1 repeatedly)" [in more human-readable form].

        You have a slight typo. Should be: "... as n goes from 1 to infinity ..."

        • Re:typo (Score:5, Funny)

          by mcrbids (148650) on Wednesday July 02, 2008 @02:47PM (#24034641) Journal

          The Riemann zeta function is \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} [written for LaTeX], or "the sum of 1/(n^s) as n goes from 0 to infinity (increasing by 1 repeatedly)" [in more human-readable form].

          You have a slight typo. Should be: "... as n goes from 1 to infinity ..."

          You have a slight typo. It should be: "You have a slight typo. It should be: ..."

      • Re: (Score:3, Funny)

        Okay...I would ask WHY this is important, but someone is ponying up a million bucks for the solution. THAT tells me this is important. I'm not sure if I care why...
        • Re:Tried to RTFA (Score:5, Informative)

          by JohnsonJohnson (524590) on Wednesday July 02, 2008 @03:00PM (#24034827)

          It's important because the zeros of the zeta function tell you how the prime numbers are distributed and prime numbers are to number theory as elements are to chemistry, everything you could care about is built out of them. The RH is also related to host of other more esoteric, but no less important conjectures; the truth of a large part of modern mathematics relies on knowing if the RH is true or false.

          Although it's unlikely to impact the storage capacity of a flash drive any time soon the zeta function shows up in high energy physics and thus does have real world consequences.

          • Re: (Score:3, Interesting)

            by PlatyPaul (690601)
            Here's another easy-to-grasp one: public key encryption (think: credit card purchases online) is dependent upon the use of large primes. Large primes are currently not the easiest/fastest to find - what if you knew better where to look for them?
      • by CarpetShark (865376) on Wednesday July 02, 2008 @02:57PM (#24034787)

        I just finally found a simple explanation of complex numbers, and just heard of this Riemann Hypothesis, so I may be way off, but let me try to put what (I think) I've figured out so far in layman's terms for the rest of the lost souls:

        Riemann was interested in the zeros to this function, where s is a complex number. He conjectured that all zeros (aside from those of the form s = -2c, where c is a positive integer) would have to be of the form (1/2) + ki, where k is a constant and i is the square root of -1.

        Basically, 10 trillian calculations have been done involving certain complex numbers, which all show a clear pattern: if you get an answer of 0, the real part of the number given to the function always seems to be 0.5. As yet, no one has proven this, and so, presumably, no one truly understands why that's the case yet. Also, presumably, when we do understand it, we'll have forward (either in a a step or a leap) in our ability to use complex numbers (and the multi-dimensional calculations they represent.

    • Numb3rs (Score:5, Funny)

      by netsavior (627338) on Wednesday July 02, 2008 @12:28PM (#24032541)
      Charles Eppes: Imagine you have an infinite number of plot holes, and you want to test how they compare to imaginary numbers. The Riemann Hypothesis states that I can use the zeros in this formula to predict how bullets will bounce off of concrete to a degree of statistical accuracy that it will actually give me the social security number of the guilty shooter.
    • Re: (Score:3, Funny)

      by Frankie70 (803801)

      It's like 50 football fields laid in line from here to Riemann.
      Rieman sounds like a place in Germany.

  • by deft (253558) on Wednesday July 02, 2008 @11:46AM (#24031779) Homepage

    Was reading wikipedia because I have no idea why this is important, but need to know enough to impress my friends (and by that I mean, alienate).

    But I noticed this is such a big deal, theres a cool million waiting for the person that proves it. John Nash in "beautiful Mind" tries to prove this one too. Sorry gladiator... not today!

    So yeah, Check it out, notice the offer at the end, after all the completely unintelligible mathematicrap:

    Riemann hypothesis

    The Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in mathematics. It has been an open question for almost 150 years, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs.

    The Riemann hypothesis (RH) is a conjecture about the distribution of the zeros of the Riemann zeta-function (s). The Riemann zeta-function is defined for all complex numbers s 1. It has zeros at the negative even integers (i.e. at s = 2, s = 4, s = 6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:

    The real part of any non-trivial zero of the Riemann zeta function is ½.
    Thus the non-trivial zeros should lie on the so-called critical line, ½ + it, where t is a real number and i is the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line.

    The Riemann hypothesis is one of the most important open problems of contemporary mathematics, mainly because a large number of deep and important other results have been proven under the condition that it holds. Most mathematicians believe the Riemann hypothesis to be true.[1] A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof.[2]

    • by rufty_tufty (888596) on Wednesday July 02, 2008 @11:54AM (#24031953) Homepage

      Good explanation here too:
      http://www.irregularwebcomic.net/1960.html [irregularwebcomic.net]

      • by nwf (25607)
        That is probably the best explanation I've seen, thanks! And it makes use of LEGO, another plus!
      • http://www.irregularwebcomic.net/1960.html

        Great! Now how am I supposed to get any work done.
    • Tough problems (Score:4, Interesting)

      by dj245 (732906) on Wednesday July 02, 2008 @11:55AM (#24031975) Homepage
      Part of the reason these problems are so tough because to solve them, you have to understand what the problem is first. I studied the Riemann hypothesis in college for a good week and I'm still not sure where you might begin solving it. Like the Navier-Stokes equations (another big problem with a big prize) solving it will probably require the invention of some new mathematics. Its not simply a matter of dividing by 3 and carrying the 2. I don't know about you but I haven't the slightest idea about how to go about inventing new math. That's the realm of Newton and Einstein, and few others.

      New math is the only way to go about solving some of these problems.
      • New math is the only way to go about solving some of these problems.

        You mean like this? [aol.com]

      • by afabbro (33948) on Wednesday July 02, 2008 @12:39PM (#24032717) Homepage

        ...solving it will probably require the invention of some new mathematics. Its not simply a matter of dividing by 3 and carrying the 2.

        If you're carrying numbers when dividing, I guess you are inventing new math :-)

        • Re: (Score:3, Funny)

          by jd (1658)
          It's easier to have just one heavy maths function and one trivial maths function than two heavy maths functions, so division is easiest implemented as multiplication with the inverse of one of the two numbers, inverses being relatively trivial in exponential notation. As only computers operate this way, the grandparent poster is obviously an artificial intelligence.
    • by Anonymous Coward on Wednesday July 02, 2008 @12:03PM (#24032131)

      The Riemann hypothesis is considered the most important unsolved problem in math. But, considering the source here (random paper on ArXiv by complete unknown), there's no real reason to believe this paper is correct. The number of incorrect proofs to major mathematics problems every year is staggering.

    • by olyar (591892)

      John Nash in "Beautiful Mind" tries to prove this one too.

      One of the things I remember from the book is that he and his wife had a running joke that all babies know the solution to this problem and then forget it when they learn to talk. Maybe Xian-Jin Li had a flashback.

      • by olyar (591892)
        Ahh - here's where it came from:

        Paul Erdos once said all babies (he used to call them epsilons, because babies are really small!) remember the solution for Riemann Hypothesis. The only problem though is that they tend to forget everything once they reach the age of six month.

        Found that here [wordpress.com]

      • by afidel (530433)
        Flashbacks to babyhood, that reminds me of another outstanding film, the directors cut to The Butterfly Effect. If you haven't seen it I highly recommend it though it's definitely not for people who cry easily or women who have lost a child!
    • Re: (Score:3, Funny)

      John Nash in "beautiful Mind" tries to prove this one too.

      And I would have succeeded if it weren't for these meddling kids! What do you mean you can't see them?!

  • Reimann? (Score:5, Funny)

    by areusche (1297613) on Wednesday July 02, 2008 @11:46AM (#24031783)
    Reimann? Like the Noodles right?
  • Hmmm.... (Score:5, Funny)

    by Otter (3800) on Wednesday July 02, 2008 @11:46AM (#24031791) Journal
    The only part of it I understood was:

    The author is grateful to J.-P. Gabardo, L. de Branges, J. Vaaler, B. Conrey, and D. Cardon who have obtained academic positions in that order for him during his difficult times of finding a job.

    Sounds about par for the course for academic hiring, and it sounds like he's still pretty traumatized from it. I hope this works out for him and he can go around flipping off all the hiring committees who turned him down.

    • Re: (Score:2, Informative)

      by Anonymous Coward

      It's brutal trying to try to get into academia in a field that doesn't produce money. The sad thing is that departments want to hire more people but there is never any money or open positions and tenured professors hang onto their positions until they die. Things are a little better in physics than math, but not much (I am an experimental physicist).

      I had an undergraduate professor tell us endlessly to NOT go into physics, as it would make us miserable careerwise. I'm still in physics, but most of my fri

      • Re:Hmmm.... (Score:5, Funny)

        by Anonymous Coward on Wednesday July 02, 2008 @01:26PM (#24033467)

        I had a history professor tell me that if he knew how hard it would be to get to where he was, he never would have been a history major.

        Well, that's all in the past now.

  • Math = $$ (Score:5, Funny)

    by RabidMoose (746680) on Wednesday July 02, 2008 @11:47AM (#24031805) Homepage
    According to the http://en.wikipedia.org/wiki/Riemann_hypothesis [wikipedia.org] wikipedia article, this means $1,000,000 if the proof turns out to be valid. Unfortunately, I didn't understand anything else in that article.
    • He gets a million because a lot of modern mathematics assumes it is true but no-one can (so far) prove it ....

      It he is correct a lot of mathematicians breathe a huge sigh of relief

      If someone proves it is false then mathematics collectively panics and a lot of proofs will have to be re-written ...

  • So what? (Score:3, Insightful)

    by feijai (898706) on Wednesday July 02, 2008 @11:47AM (#24031815)
    arXiv has become the repository for junk that couldn't pass peer review. Wake me up when we see a published journal article.
    • Also, the proof of something that complicated is likely so complicated that only the very best minds would even be able to prove that the proof was wrong.

    • Re:So what? (Score:5, Informative)

      by JambisJubilee (784493) on Wednesday July 02, 2008 @12:14PM (#24032335)

      I think you misunderstand the scope and purpose of arXiv. arXiv is a repository for *preprints*.

      By uploading the file to arXiv before submitting it, not only do you ensure that those that can't afford $10,000+ subscription fees can access the article, but you open up your findings to a much wider international audience.

      The lack of peer review is not necessarily a liability in this situation

  • not so fast (Score:5, Informative)

    by Anonymous Coward on Wednesday July 02, 2008 @11:49AM (#24031845)

    there are "proofs" of the Riemann hypothesis on the arXiv every few weeks. Don't believe it 'til it's vetted.

    • by Anonymous Coward on Wednesday July 02, 2008 @12:07PM (#24032185)

      Yeah. arXiv once published my paper that shows cases where P = NP; I proved it conclusively for the cases where P = 0 and/or N = 1, but so far I haven't gotten my $1,000,000.00 check from the Clay Math Institute.

    • Re: (Score:2, Interesting)

      Indeed. Among some mathematicians it is a pleasant diversion to take bets on which of the major unsolved (or unprovable) problems has the most solutions appear on the arXiv this week.
    • Re: (Score:3, Interesting)

      That's true, but most of them are obvious drivel. I have looked through this one, and it is clearly a real attempt by a genuine mathematician who understands the relevant background. I'd still bet on it being wrong, but not stupidly wrong.
  • Dolly Parton was 69 lbs over weight. The doctor said that's 222 much! You need to lose 51 x 8 days. That left her:

    6922251x8=55378008

  • Oblig. (Score:5, Funny)

    by JuanCarlosII (1086993) on Wednesday July 02, 2008 @11:56AM (#24031983)
  • Ok, so many have tried, all have failed.

    It may take a decade to test the assertions that this so called proof attempts to demonstrate.

    Perhaps we could give the guy a consolation prize, wait for the work to be "proven" wrong and then off course, issue an Apology:

    http://www.math.purdue.edu/~branges/apology.pdf [purdue.edu] :-)

    -Hack

    PS: Does anyone find it STRANGE that the guy who can solve this problem has issues finding a job?

    WTF?

    • Re: (Score:2, Interesting)

      by JuanCarlosII (1086993)
      Not really, the kind of person who would solve a problem of this nature is probably going to be the Andrew Wiles reclusive genius type - a lot like the Russian gent whose name escapes me who solved the Poincare Conjecture. Thus he's not necessarily going to be too keen to teach/lecture/supervise and so would possibly not be too attractive to prospective employers.

      I doubt too many Maths faculties in the world have people working full-time on the Riemann Hypotheses.

      Of course I echo your sentiments that
    • by 1729 (581437)

      Interestingly, DeBranges was Xian-Jin Li's advisor:

      http://www.genealogy.math.ndsu.nodak.edu/id.php?id=16641 [nodak.edu]

  • by vingilot (218702)

    hEll

  • That reminds [explosm.net] me.
  • This guys advisor, according to the Math Genealogy Project, is Louis deBranges. DeBranges also claimed to have proven this a few years back, but his proof was not accepted (for reasons unknown to me). The $1M might still be safe.
  • Simple Simon actually met a Riemann, after all?!

    I thought that was just a hypothesis!

  • by Anonymous Coward on Wednesday July 02, 2008 @01:13PM (#24033239)

    Section two of the wiki article (http://en.wikipedia.org/wiki/Riemann_hypothesis) is the great importance here. If indeed there is a proof of Riemann's Hypothesis, then there is a similar proof of the Generalized Riemann Hypothesis, which is in turn a big step in finding the exact distribution of prime numbers.

    Finding the distribution of prime numbers has epic consequences, like breaking most encryption, for starters.

    • by payola (1127685) on Wednesday July 02, 2008 @05:26PM (#24036561)
      The Riemann Hypothesis and RSA encryption both have to do with prime numbers, but the relationship between the two pretty much ends there. To break RSA you need to know how to factor large numbers quickly. RH, on the other hand, pretains to the distribution of prime numbers. It's pretty unlikely that a proof would make computers any faster at factorizing.

      So this begs the question that a lot of people have been asking on this thread: why should you care? There tongue-in-cheek answer is that a solution is worth $1,000,000. While that response may suffice for non-mathematicians, mathematicians would have another, more important reason to celebrate. RH and its generalization, the Grand Riemann Hypothesis, have an absolutely enormous number of profound impliations in number theory, and it is difficult to overstate how critical a proof of either would be. (The implications are too technical to write about here, but you can read about them in most good survey books on analytic number theory; for example, see section 5.8 of Iwaniec & Kowalski [amazon.com]). A successful proof probably won't affect your life in any meaningful way (unless you work with analytic number theory for a living), but it would be monumental in the world of math - indeed, this is precisely why there's a reward for solving it. If that's not enough for you, just remember that many mathematicians are motivated not by fame or money but by the beauty and elegance of mathematics, and any proof of RH would establish a truly beautiful and amazing result.

      Of course, there's also the question: is Li's proof correct? I certainily don't know, and I doubt anyone will for quite some time, but there's an interesting story. Li's Ph.D. adviser was Louis de Branges [nodak.edu] who, as noted on this very website [slashdot.org], claimed to prove RH in 2004. His proof has not been accepted by the mathematical community and is widely considered to be incorrect, in large part because the method he wclaims to use was shown, in a 2000 paper [arxiv.org] co-authored by none other than Xian-Jin Li, to have holes in it.
  • by Anonymous Coward

    I can't believe they are brazenly going forward with research into this subject without knowing if it could possibly lead to the creation of a black hole that will swallow the earth.

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