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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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  • by InvisblePinkUnicorn (1126837) on Wednesday July 02, 2008 @11:38AM (#24031635)
    37047734
  • 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]

  • 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 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 Mr. Sketch (111112) <mister,sketch&gmail,com> on Wednesday July 02, 2008 @12:12PM (#24032291)

    I briefly looked through the proof, and it only claims to be a proof in the rational number field, not for all real numbers. It's still a step in the right direction, but not a full proof.

  • 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

  • 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].
  • Re:Hmmm.... (Score:2, Informative)

    by Anonymous Coward on Wednesday July 02, 2008 @12:24PM (#24032469)

    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 friends are not, and I totally understand his point now. 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.

  • 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 ..."

  • by Anonymous Coward on Wednesday July 02, 2008 @12:45PM (#24032805)

    No. Every number field has its own zeta function. The standard Riemann hypothesis concerns that of the rationals.

  • by Cowculator (513725) on Wednesday July 02, 2008 @01:39PM (#24033653) Homepage
    No, you're wrong because you have no idea what you're talking about. Every number field has its own zeta function which roughly describes the distribution of prime ideals in that field, and the Riemann zeta function is the one corresponding to the rational field. The Riemann hypothesis states that the Riemann zeta function (that is, the one for the field of rational numbers) has no zeros whatsover, rational or otherwise, on the critical strip 0 < Re(s) < 1 except along the line Re(s) = 1/2, and this is exactly the statement he's claiming to have proved.
  • Re:Dirty Words (Score:2, Informative)

    by Anonymous Coward on Wednesday July 02, 2008 @01:58PM (#24033953)
    Hold the calculator upside down, mod-who-didn't-get-the-joke...
  • 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.

  • 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:Tough problems (Score:3, Informative)

    by AshtangiMan (684031) on Wednesday July 02, 2008 @03:36PM (#24035231)
    Thank god (or Goedle) that they don't have to be both consistent and complete.
  • by Anonymous Coward on Wednesday July 02, 2008 @05:26PM (#24036553)

    Ahem, Xian-Jin Li has a mathematical criterion named after him: http://en.wikipedia.org/wiki/Li%27s_criterion

  • 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.
  • Re:Tried to RTFA (Score:2, Informative)

    by Garridan (597129) on Wednesday July 02, 2008 @10:21PM (#24039143)
    JSMath.
  • by payola (1127685) on Thursday July 03, 2008 @04:12PM (#24050061)
    Extremely highly-regarded mathematician Terence Tao [wikipedia.org] has said, in response to a comment left on his blog [wordpress.com], that the proof is probably incorrect.
  • by Anonymous Coward on Thursday July 03, 2008 @05:14PM (#24050951)

    Fields medal winner, and expert in the field,
    Alain Connes sees a problem with the attempted
      proof.

    http://noncommutativegeometry.blogspot.com/2008/06/fun-day-two.html?showComment=1215071400000#c8876982000013974667

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