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The Almighty Buck Science

Mathematician Claims Proof of Riemann Hypothesis 561

Posted by timothy
from the peer-review-pending dept.
TheSync points to this press release about a Purdue University mathematician, Louis de Branges de Bourcia, who claims to have "proven the Riemann hypothesis, considered to be the greatest unsolved problem in mathematics. It states that all non-trivial zeros of the zeta function lie on the line 1/2 + it as t ranges over the real numbers. You can read his proof here. The Clay Mathematics Institute offers a $1 million prize to the first prover."
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Mathematician Claims Proof of Riemann Hypothesis

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  • by Anonymous Coward on Wednesday June 09, 2004 @06:01PM (#9382541)
    It's that mathematicians love to exaggerate! Like infinity is infinite, or pi goes on forever! Those guys are always talking big.
    • by Anonymous Coward on Wednesday June 09, 2004 @07:01PM (#9382987)
      mmmmmmmm......infinite pie..!
    • Degrees of infinity... now that screwed with my head...

      Q.

    • by Ckwop (707653) * <Simon.Johnson@gmail.com> on Thursday June 10, 2004 @02:25AM (#9384830) Homepage

      De branges is a bit of a crank on the Riemann hypothesis. No-one believes his approach(s) will work. This is well documented in the book "Riemann's Zeros". When some of the leading mathematians were asked about his approach they said it was "full of errors" and "unlikely to work". The only reason he is given the light of day is because he managed to prove to the Bieberbach conjecture. That was a difficult problem, hats off to him for getting it aswell, but it's no Riemann hypothesis!

      Rest assured, we'll all be dead and burried when it actually gets solved.

      Simon

      • by smallfries (601545) on Thursday June 10, 2004 @06:19AM (#9385506) Homepage
        It would appear that mathworld.com agrees with you...

        ----------------

        Riemann Hypothesis "Proof" Much Ado About Noithing
        A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing.
  • Apology (Score:5, Funny)

    by Anonymous Coward on Wednesday June 09, 2004 @06:02PM (#9382552)
    Apology for the proof of the Riemann hypothesis (in pdf format). [purdue.edu]

    "We humbly apologize for the complete illegibility of this proof. The mathematician responsible has been sacked."
    • by Anonymous Coward
      Karma-whoring free [216.239.39.104].
    • Re:Apology (Score:5, Funny)

      by Tackhead (54550) on Wednesday June 09, 2004 @06:08PM (#9382601)
      > "We humbly apologize for the complete illegibility of this proof. The mathematician responsible has been sacked."

      "A Slashdotter has discovered a truly wonderful proof of the sacking of the mathematician responsible, but his bandwidth is too narrow to host it!"

    • WTF? Mods? (Score:5, Informative)

      by Unnngh! (731758) on Wednesday June 09, 2004 @06:10PM (#9382613)
      From reference.dictionary.com:

      Apology - 2: a formal written defense of something you believe in strongly

      This should at most have earned a "Funny", or is there something I'm missing here?

      • by thefinite (563510) on Wednesday June 09, 2004 @06:37PM (#9382826)
        This should at most have earned a "Funny", or is there something I'm missing here?

        Yeah, I think you missed:
        Equivocation - \E*quiv`o*ca"tion\, n. The use of expressions susceptible of a double signification, with a purpose to mislead boneheaded moderators, especially when you are just making a joke.

    • Re:Apology (Score:4, Insightful)

      by badboy_tw2002 (524611) on Wednesday June 09, 2004 @06:10PM (#9382614)
      Uh, the above comment was a joke people. The quote in the parent post does NOT appear in the document. Apology in this case means a defense of the proof.
    • Re:Apology (Score:4, Informative)

      by ssssmemyself (709098) on Wednesday June 09, 2004 @06:13PM (#9382635) Homepage
      Note to mods: Mod parent funny, not interesting! This is a play off a quote from the beginning credits sequence in Monty Python and the Holy Grail. As for the pdf link, it's the first link in the purdue page referenced in the article. RTFA, people!
      • Re:Apology (Score:5, Interesting)

        by MerlynEmrys67 (583469) on Wednesday June 09, 2004 @06:50PM (#9382921)
        Of course if I were to RTFA - and more importantly UTFA (Understand the Article) I wouldn't be able to post this for another 2 years or so...

        As it is, it looks like he proposed this solution over a year ago and has been getting it vetted in a tightly controlled community. Now that the cat is out of the bag he will have to get it into a peer reviewed journal (takes 6 months or so) and wait 2 years to see how it is bashed...

        Yeah - that is about the time it would take for me to UTFA, except I am not a Mathemetician, so add in another 6-8 years to get that training as well. So I will get back to you sometime around 2120 with an insightful comment after UTFA

    • Re:Apology (Score:5, Interesting)

      by gniv (600835) on Wednesday June 09, 2004 @06:51PM (#9382925)
      The last paragraph of the article is interesting:
      A curious coincidence needs to be mentioned as part of the chain of events which con-
      cluded in the proof of the Riemann hypothesis. The feudal family de Branges originates in
      a crusader who died in 1199 leaving an emblem of three swords hanging over three coins,
      surmounted by the traditional crown designating a count, and inscribed with the motto
      "Nec vi nec numero." This is a citation from Chapter 4, Verse 6, of the Book of Zechariah:
      "Not by might, nor by power, but by my Spirit, says the Lord of Hosts." The chateau de
      Branges was destroyed in 1478 by the army of Louix XI of France during an unsuccessful
      campaign to wrest Franche-Comte from the heirs of Charles the Bold of Burgundy. The
      family de Branges performed administrative, legal, and religious functions in Saint-Amour
      for the marquisat d'Andelot during Spanish rule of Franche-Comte. Francois de Branges
      of Saint-Amour received the seigneurie de Bourcia in 1679 when Franche-Comte became
      part of France. The chateau de Bourcia remained the home of his descendants until it was
      destroyed by Parisian revolutionaries in 1791. The chateau d'Andelot near Saint-Amour,
      which survived the revolution, was bought in 1926 by Pierre du Pont, an elder brother
      of Irenee du Pont, for a nephew assigned in diplomatic service to France. This coinci-
      dence accounts for the interest which Irenee du Pont showed in a student of mathematics.
      The ruin of the chateau de Bourcia overlooks a fertile valley surrounded by wooded hills.
      The site is ideal for a mathematical research institute. The restoration of the chateau for
      that purpose would be an appropriate use of the million dollars offered for a proof of the
      Riemann hypothesis.
      That's quite noble of him.
      • Re:Apology (Score:4, Funny)

        by Ralph Wiggam (22354) on Wednesday June 09, 2004 @07:53PM (#9383254) Homepage
        The Bourcia Mathematical Research Institute will involve more whores and cocaine than a typical research institute, but for tax purposes it's a research institute.

        -B
      • Re:Apology (Score:5, Interesting)

        by dasmegabyte (267018) <das@OHNOWHATSTHISdasmegabyte.org> on Wednesday June 09, 2004 @10:41PM (#9384002) Homepage Journal
        This guy is an all around class act. I've always found mathematicians to be kind of standoffish, and while this guy is obviously at the top of his field, he's also on top of the rhetorical game, the very structure of this "Apology" shows that he's having a great deal of fun with his chosen profession.

        My favorite selection:
        The solution of a celebrated problem creates a disturbance in the otherwise quiet flow of mathematical events. The solution escapes the planning of committees. Colleagues are unprepared because the possibility of a solution has not been included in their research proposals. Students have avoided related thesis topics because of the risk that the work will not be welcome to a prospective employer. Friends are discouraged from research activity by the demands of the situation created by the solution. The manuscript, which is necessarily written at the highest research level, is readable only to a limited audience. An introduction is therefore needed which makes available the opportunities created by the solution. This is done by supplying motivation for the argument in a chronological order which also gives an account of how the solution was obtained.

        Hilarious stuff. He apologizes to the people who will now feel the need to go over his proof with a fine toother comb, looking for mistakes...and also explains (three pages in) why he's chosen to start his proof with a history of the golden age of mathematics, stretching back to Newton. Basically, he's saying "oh hey, thanks for joining me. I was just explaining ALL OF MATHEMATICS for those playing at home. Bear with me, this one's worth it, and I promise you can get back to your euclidian algorithms and Ving diagrams in short time."

        Ever read "The Life and Opinions of Tristram Shandy?" It's an amazing book from the 18th century, which attempts to tell a simple narrative but due to the extremely schizophrenic style of the narrator, it keeps breaking down into tangential pockets of narrative self awareness. Basically, the author wrote from the perception of a disturbed dandy who couldn't keep his mind on the task at hand, an author who keeps apologizing to his readers for the inconvenience of his own poor editing.

        This mathematical proof reminds me a lot of this book...the text of the proof doesn't act as though the proof isn't something interesting or ground breaking, nor does it make a big deal of this. It just ambles on in all directions until the Riemann hypothesis is well and truly proven, but with no real hurry to illustrate the proof until the outlines have been inked. Not that I know for sure that Riemann is proven or isn't...my brain was full when I got to differentials. But if it is, this paper will stand out not only as a great work of mathematics, but a great work of WRITING about mathematics.

        I'm going to read it again. Maybe I'll understand it this time!
        • Well, not exactly (Score:4, Informative)

          by mrgeometry (689087) on Thursday June 10, 2004 @09:00AM (#9386449)
          Sorry, but...

          It is not proved; he is not at the top of his field; this "paper" will be quickly forgotten among professional mathematicians; and I doubt any professional mathematician is going over the proof with any sort of comb.

          L. de Branges first achieved fame for proving the Bieberbach conjecture [wikipedia.org]. His proof went through strange and abstract methods. He went on the road to present his proof at various seminars in France, Russia, etc; IIRC a bunch of Russian students got very excited and basically rewrote his proof. Their new proof was much shorter and avoided the use of strange methods. Nowadays, their proof is remembered and his is not, but the proof still bears his name, since after all he was the first to come up with *some* kind of proof, and their proof did more or less come out of his.

          So he deserves credit for that, and it was quite an achievement to prove the Bieberbach conjecture. But even then he was using unwieldy proofs with unnecessarily abstract methods.

          For many years he has been claiming to have a proof of the Riemann Hypothesis. Professional mathematicians stopped listening a long time ago.

          This guy is washed-up.

          I whole-heartedly agree that this short article is hilarious, but I would like to add the adjective condescending. What kind of asshole apologizes for solving a problem? Does he think he lives on some higher plane, and therefore must take direct, personal responsibility for every aspect of our lives?

          Look at how G. Perelman submitted his ideas on proving the Poincare conjecture [wikipedia.org] just a little while ago. He didn't waste anyone's time by rehashing the already-available history of the problem or its wider context in mathematics. Nor did he apologize for having an idea. Rather, he submitted his ideas for consideration, with the full awareness that there may have been a mistake. .... Now, this is where I admit that I do not really understand that area of math, and have not been closely following the status of (alliteration alert) Perelman's proposed proof. Still, Perelman is a real mathematician, and even if the proof is (was?) wrong, it has real ideas of value in it.

          de Branges is so full of crap, it makes me sick.

          zach
    • Re:Apology (Score:3, Funny)

      by letxa2000 (215841)
      You know, I had this exact same idea several years ago but I figured it couldn't possibly be [b]that[/b] obvious so I figured I was just wrong. Rats. :)

  • Good job (Score:5, Funny)

    by Thinkit4 (745166) * on Wednesday June 09, 2004 @06:04PM (#9382560)
    It's too bad that most of society does not recognize truly great achievements like this. I, for one, admit interest but not enough knowledge of the details to read and understand the proof. I'm sure most people here on /., as representatives of the intelligent future of sentient life, have the interest as well.
  • by foidulus (743482) * on Wednesday June 09, 2004 @06:05PM (#9382570)
    They really should make mathematics more like pokemon, it would get more people interested in the subject
    Riemann-chu, I prove you! Then bust out the paper.
    • by Felinoid (16872) on Wednesday June 09, 2004 @06:19PM (#9382689) Homepage Journal
      Mathomon?
      Yeah but then a few years later Yu-Physics-Oh comes along and replaces it in popularity. Then before you know it that two is gone replaced by annother populare science.
      Plus it would replace Arceology the gathering.

      Magic The Gathering, Pokemon and yugioh are in the 15 minuts of fame catagory. Populare today gone tomarow.

      I don't want Math to be gone tomarow. I'm counting on it to stay for a while.

      Now english I wouldn't mind if it's own end was spelled out. You can see the proof reading this very post.
  • by ajboyle (547708) on Wednesday June 09, 2004 @06:06PM (#9382576)
    I read through his proof and...nope, it's wrong. I know the real answer, but am leaving it as an exercise for the interested student.
  • Failed proof (Score:5, Informative)

    by MobyDisk (75490) on Wednesday June 09, 2004 @06:06PM (#9382584) Homepage

    Ha! They've already found an error in the proof! All that he posted was his apology! [purdue.edu] :-)

    Yes, I was actually confused at first. For the non-math geeks like myself, who are feeling stupid, look at definition 2a of apology [reference.com].
  • by Anonymous Coward on Wednesday June 09, 2004 @06:07PM (#9382595)
    I don't want to give it away, but you'll see it.
  • by kaalamaadan (639250) on Wednesday June 09, 2004 @06:08PM (#9382596) Journal

    "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"

    David Hilbert

    • by Mark_in_Brazil (537925) on Wednesday June 09, 2004 @06:41PM (#9382867)
      Hilbert may have been referring to the importance of the Riemann Conjecture, and not the difficulty of proving it.

      Really, folks, this is a big deal if it's true. It just doesn't get the attention Fermat's Last Theorem did because it's harder to understand what it means and why it's important.

      After all, most people don't even know what complex numbers are, much less complex functions. The zeta function, then, is already beyond the understanding of most people, not because they're incapable, but because they're not interested. But the implications of the Riemann Conjecture are far-reaching indeed, affecting things like quantum mechanics and statistical physics.

      --Mark
      • "The structure of mathematical journals creates the impression that mathematics is fragmented into unrelated disciplines. The underlying unity of mathematics is however maintained by problems which span these disciplines. ... The Riemann hypothesis is listed as an important link between algebra and analysis."

        The significance may be more in the mathematical machinery required to prove it than in the result itself.
    • by rattler14 (459782) on Wednesday June 09, 2004 @06:54PM (#9382945)
      "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"

      Oh yeah? Mine would be "Is Doom 3 out yet?"

      Honestly, which is more likely?
  • by martinX (672498) on Wednesday June 09, 2004 @06:08PM (#9382598)
    You know you're in trouble when you don't even understand the question.
    • by SamSim (630795) on Wednesday June 09, 2004 @08:20PM (#9383356) Homepage Journal

      First: complex numbers [wolfram.com], explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i [wolfram.com]. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane [wolfram.com]. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

      Right. Now the Riemann Zeta Function [wolfram.com] is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

      Now, a zero [wolfram.com] of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

      As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis [wolfram.com] suggests that they are... but until today nobody has been able to prove it.

      • by Dasein (6110) * <tedc@NOSpam.codebig.com> on Wednesday June 09, 2004 @11:08PM (#9384116) Homepage Journal
        There's the occasional post that deserves to be modded to "+10 -- Best Damn Thing I've Read On Slashdot This Year".

        Thanks!
  • Is it... (Score:5, Funny)

    by Anonymous Coward on Wednesday June 09, 2004 @06:08PM (#9382602)
    ... 42?
  • Impact on crypto? (Score:4, Interesting)

    by Anonymous Coward on Wednesday June 09, 2004 @06:11PM (#9382624)
    This theorem is a theory of how prime numbers are distributed...so does it's proof have any impact on crypto? Does it make it any easier to find prime numbers?
    • by Anonymous Coward on Wednesday June 09, 2004 @06:16PM (#9382664)
      The Riemann Hypothesis, among other things, implies that the Prime Number Theorem is off in the distribution of primes by no more than O(sqrt(n)*log(n)). However even without the full result, we already had very good error bounds for the approximation of the prime number theorem for "small" numbers, including numbers far larger than any which come up in cryptography.
    • Re:Impact on crypto? (Score:3, Informative)

      by Unnngh! (731758)
      I don't know nearly enough math to understand the proof, but judging that the hypothesis was made by Reimann quite a while ago, and this is a proof of that hypothesis, I would conclude that the theory has been extant but unsubstantiated until now.

      So, in short, no, no help for cracking crypto based on primes...though the article does mention possible crypto applications down the line. I'm not sure what, exactly, those would be.

    • Re:Impact on crypto? (Score:5, Informative)

      by susano_otter (123650) on Wednesday June 09, 2004 @06:37PM (#9382824) Homepage
      This theorem is a theory of how prime numbers are distributed...

      It's actually a little more complex than that.

      Riemann was investigating the distribution of prime numbers. Along the way he devised (discovered?) the Zeta Function, which describes with considerable accuracy the distribution of prime numbers. While working with the Zeta Function, he discovered an interesting property: It appeared that all the non-trivial zeroes of the function had a real part of one-half. However, since this property of the function was not related to the prime-distribution work he was doing, he did not bother to prove this apparent property, which came to be known as the "Riemann Hypothesis" (presumably, once it is proven it will be known as the Riemann Theorem, or some such).

      Thus, the Riemann Hypothesis is in fact tangential to (and possibly unrelated to) the distribution of prime numbers. Riemann's notes on the Zeta Function, regarding his work on prime distribution, are pretty explicit about this.

      • by onemorehour (162028) on Wednesday June 09, 2004 @07:33PM (#9383158)
        It's actually a little more complex than that.

        *smack!*

      • Re:Impact on crypto? (Score:5, Informative)

        by NonSequor (230139) on Wednesday June 09, 2004 @08:46PM (#9383460) Journal
        Along the way he devised (discovered?) the Zeta Function, which describes with considerable accuracy the distribution of prime numbers.


        Actually, as with most things Euler was the first to study it. The zeta function is also the simplest of a class of functions that Dirichlet studied Dirichlet L-series. There is also a Generalized Riemann Hypothesis that states that no Dirichlet L-series has zero with real part greater than 1/2.

        The Riemann Hypothesis is more than tangential to the study of the distribution of primes. There is a function derived from the distribution of the primes that can be expressed in terms of the non-trivial zeros of the zeta function. The Prime Number Theorem is also equivalent to the statement that the zeta function has no zeros with real part 1. The Generalized Riemann Hypothesis implies the weak form of Goldbach's conjecture (i.e. that any odd number greater than 7 can be expressed as the sum of three odd primes).
    • Re:Impact on crypto? (Score:5, Informative)

      by cperciva (102828) on Wednesday June 09, 2004 @06:39PM (#9382841) Homepage
      does it's proof have any impact on crypto?

      No. Almost all mathematicians have assumed for years that GRH is true anyway; proving it would mean that all those footnotes ([1] Under the assumption of the Riemann Hypothesis) could be removed, but that's the only practical effect it would have.
  • Does this affect prime based public key schemes at all? How does it affect them?

    • by Anonymous Coward on Wednesday June 09, 2004 @06:33PM (#9382793)
      Nope, probably not.
      Most mathematicians felt that the Riemann Hypothesis was true so that this view has been taken into consideration into mathematics for a long time. Perhaps if he developed some new methods for playing with numbers in the proof, but it doesn't seem like it to me.
      There's a ton of math papers that begin with "Assume the extended riemann hypothesis...".

      At least that's my guess.
  • by ak_hepcat (468765) <<leif> <at> <denali.net>> on Wednesday June 09, 2004 @06:18PM (#9382681) Homepage Journal
    I knew it was a hoax when he started discussing his Paley-Wiener space...
  • by TorKlingberg (599697) on Wednesday June 09, 2004 @06:19PM (#9382682)

    Will the media keep publishing claims of extraordinary mathematical findings without checking the facts forever?

    Just like this one over again:
    Swedish Student Partly Solves 16th Hilbert Problem [slashdot.org]

    That's what I like about /. If the article is wrong, there is always the comments there to solve it.

    • The media just report the facts (insert joke here), and the fact -- in this case -- is that someone claims to have made an extraordinary mathematical discovery. Therefore, in this case, we are the fact-checkers. Or, rather, anyone who understands enough math to sift through 124 pages of an alleged proof (to prove the proof?) are the fact-checkers.
  • by Anonymous Coward on Wednesday June 09, 2004 @06:20PM (#9382693)
    Although I hope de Branges has found a proof, I'm not too optimistic. It seems that de Branges has a reputation among mathematicians for going off half-cocked. He does have the Bieberbach proof under his belt, though, so you never know.
  • A Proof .... Maybe (Score:4, Interesting)

    by BrownDwarf (615206) on Wednesday June 09, 2004 @06:20PM (#9382697)
    It seems that the proof hasn't been reviewed yet. He may have it -- but lots of good folks have tried, without success. This from Science Daily: http://www.math.purdue.edu/~branges/ . While mathematicians ordinarily announce their work at formal conferences or in scientific journals, the spirited competition to prove the hypothesis - which carries a $1 million prize for whomever accomplishes it first - has encouraged de Branges to announce his work as soon as it was completed. "I invite other mathematicians to examine my efforts," said de Branges, who is the Edward C. Elliott Distinguished Professor of Mathematics in Purdue's School of Science. "While I will eventually submit my proof for formal publication, due to the circumstances I felt it necessary to post the work on the Internet immediately."
  • quick google search (Score:3, Interesting)

    by cancerward (103910) on Wednesday June 09, 2004 @06:24PM (#9382723) Journal
    ... shows [google.com] that he's been offering "proofs" since July 1989. I see from MathSciNet [ams.org] that he has 87 papers from 1958 to 1994, but isn't this a bit like the boy who cried wolf?
    • by Lane.exe (672783) on Wednesday June 09, 2004 @06:35PM (#9382810) Homepage
      Not really. It means he's a prolific member of the community who is not afraid to take risks with his work. Consider an experimental scientist -- in an experiment, one that turns back negative results, or on that fails, still produces important data. Similarly, this is like "experimental mathematics." If he fails, then we'll know why he fails, how far he got doing things right and other things which can point us to the correct proof.

  • Hm (Score:5, Funny)

    by blitzoid (618964) on Wednesday June 09, 2004 @06:27PM (#9382741) Homepage
    I think I speak for all non-mathematicians when I say:

    what?
  • by k98sven (324383) on Wednesday June 09, 2004 @06:28PM (#9382760) Journal
    Sorry to burst the bubble, but some usenetting shows:

    The same guy claimed [google.com] to have solved the same problem at least 4 years ago.
    The guy has a reputation [google.com] for sometimes getting it wrong.

    (Probably because he has published flawed proofs [google.com] of other well-known problems.)

    He could be right, but I wouldn't get my hopes up.
    • by roll_w.it (317514) on Wednesday June 09, 2004 @06:49PM (#9382911)
      otoh, he proved the Bieberbach conjecture in 84 and has been working on this since. Perhaps this is why he posted it before it is formally published in a journal.
    • by mrthoughtful (466814) on Wednesday June 09, 2004 @06:56PM (#9382957) Journal
      Well, he is reliably credited with solving the Bieberbach conjecture - the guy isn't a complete nut.

      However, a quick scan suggests that if his proof is indeed verified, it won't do what a lot of people want it to do: Quote from the article: "The proof of the Riemann hypothesis verifies a positivity condition only for those Dirichlet zeta functions which are associated with nonprincipal real characters. The classical zeta function does not satisfy a positivity condition since the condition is not compatible with the singularity of the function. But a weaker condition is satisfied which has the desired implication for zeros."

      So I may be wrong, but it looks like he may have found ground on a restricted interpretation of the GRH (or Generalized Riemann Hypothesis), -ie concerning Dirichlet zeta functions which are associated with nonprincipal real characters only.

      As for consequences, If GRH is indeed true, then e.g. the Miller-Rabin primality test is guaranteed to run in polynomial time.
  • The Problem (Score:5, Informative)

    by Anonymous Coward on Wednesday June 09, 2004 @06:31PM (#9382788)
    The problem is simple enough to understand, assuming you know some math basics. As most of you know, any function f(X) where f(Xo)=0 is said to have a zero at Xo. For functions of complex numbers f(z) where z=x+iy and x,y are real numbers, you obviously have the function taking on different values for every x and y, so the zeros can be anywhere on the x-y plane. For the zeta function, "trivial zeros" occur at the negative even integers (z=-2+i0,-4+i0,...) and also at points on the line x=1/2 (i.e 1/2 +iy for certain y).The Riemann Hypothesis says that all zeros that aren't negative even integers lie on this line.

    Most of you have who have taken basic calculus courses have probably seen a simplified definition of the zeta function for real intergers greater than 1. when z=n, a natural number, the zeta function reduces to the infinite series Zeta(n)= SUM (k=1-->inf) 1/k^n
  • by exp(pi*sqrt(163)) (613870) on Wednesday June 09, 2004 @06:41PM (#9382856) Journal
    I love this sentence from the article:
    The origins of the hypothesis date back to 1859, when mathematician Bernhard Riemann came up with a theory about how prime numbers were distributed, but he died in 1866, before he could conclusively prove it.
    As he didn't prove the result, either before or after his death, how can it be said that he died before he proved it? Maybe the lives of great mathematicians form arcs in some abstract space that can be extrapolated beyond their death?

    I think I might as well write my epitaph now:

    Here lies exp(pi*sqrt(163))

    He died before he could get laid by Charlize Theron
  • actual paper (Score:5, Informative)

    by Anonymous Coward on Wednesday June 09, 2004 @06:48PM (#9382909)
    The 23 page "apology" is not the actual purported proof, contrary to what the article and press release say. The actual proof is the manuscript "Riemann zeta functions", the third link on de Branges' home page, which weighs in at 124 pages!

    So if his "proof" isn't obviously wrong, it'll likely take quite a while for the experts to verify.
    • by Scorillo47 (752445) on Thursday June 10, 2004 @02:18AM (#9384802)
      The proof (or, better said, the sketch of the proof) actually starts at the end of page 21, very close to the last page. The original work is actually pretty hard to find since it is buried in so many unrelated side notes.

      Here is the general outline:
      1) At the end of page 19 he mentions that "The positivity condition which is introduced implies the Riemann hypothesis if it applies to Dirichlet zeta functions."
      2) After some introduction of the quantum gamma functions that lasts two pages, the actual proof starts at the end of page 21 with the phrase "A quantum gamma function is obtained when is nonnegative. A proof of positivity is given from properties of the Laplace transformation."
      3) The proof ends in the middle of page 23 with the a verification that W(z) is a quantum gamma function with quantum q = exp(-2*pi), obtained from a spectral theory of the shift operator.

      Overall this is just a very brief sketch of the whole proof.

      BTW, to add gas on fire, here is an exceprt from mathworld.com, which surprisingly was missed by /. until now :-)

      http://mathworld.wolfram.com

      Riemann Hypothesis "Proof" Much Ado About Noithing (sic)
      A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing.

      The counterexample to Brangles approach can be reached here: http://arxiv.org/abs/math.NT/9812166
  • Maybe (Score:3, Interesting)

    by Phragmen-Lindelof (246056) on Wednesday June 09, 2004 @06:56PM (#9382959)
    I looked at de Branges' "Apology for the proof of the Riemann hypothesis" and found no proof. Perhaps the proof is in another document?
    Even though he is a kook, I root for him; no one believed him when he claimed he had proven the Bieberbach conjecture. I believe, however, that he has claimed to have proven the Riemann hypothesis previously. One should check carefully before trusting his claim.
  • It's 42.

    Besides, I think he forgot to carry the one.
  • by jim3e8 (458859) on Wednesday June 09, 2004 @07:38PM (#9383184) Homepage
    The 23-page "Apology" referred to in the press release is also apparently mentioned in this 1996 Usenet post [google.com]. So is there a new proof? No one seems to know yet.
  • by larry2k (592744) <larry2k@mac.com> on Wednesday June 09, 2004 @07:41PM (#9383199) Homepage
    I have another proof Of Riemann Hypothesis but this text area is too small for it, anyway /. comments doesn't allow math symbols.
  • by Anonymous Coward on Wednesday June 09, 2004 @07:45PM (#9383218)
    A cool overview of why this is such an interesting hypothesis [ex.ac.uk].

    If nothing else check out the animation [ex.ac.uk].

    mind-boggling
  • by Anonymous Coward on Wednesday June 09, 2004 @09:21PM (#9383642)
    Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics [amazon.com]

    This is a very informative book about Riemann's work on the hypothesis, as well as the work of many other mathematicians. You probably need a solid college-level understanding of math to follow most of the technical explanations, but the historical parts of the book are very interesting.
  • by This is outrageous! (745631) on Wednesday June 09, 2004 @10:30PM (#9383955)
    As others mentioned, de Branges has been claiming a proof along the same lines for years. He's hard to dismiss because he actually proved the Bieberbach conjecture -- a startling exception in the series of wrong proofs he's been famous for, before and since.

    The reasons why most specialists doubt that his approach can ever yield the result are well described in this paper [arxiv.org] from 1998:

    In this note, we shall (...) give examples showing that de Branges' positivity conditions, which imply the generalized Riemann hypothesis, are not satisfied by defining functions of reproducing kernel Hilbert spaces associated with the Riemann zeta function zeta(s)
    (i.e., despite the name, the "generalized RH" proved by de Branges actually did not include the standard RH as a special case.)
  • by qpacberty (763389) on Thursday June 10, 2004 @12:47AM (#9384468)
    Berkeley Groks [groks.net] has an interview [berkeley.edu] that aired today with John Derbyshire [olimu.com] discussing the Riemann Hypothesis. He states that after talking with many mathematicians in the field, the prospects for a solution any time soon are quite low.
  • Probably a hoax: (Score:4, Informative)

    by usmcpanzer (538447) <<moc.liamtoh> <ta> <reznapcmsu>> on Thursday June 10, 2004 @01:43AM (#9384687) Homepage
    mathworld.wolfram.com
    Riemann Hypothesis "Proof" Much Ado About Noithing A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing
  • Overview of proof? (Score:3, Interesting)

    by alex_tibbles (754541) on Thursday June 10, 2004 @03:51AM (#9385110) Journal
    Could someone capable in the apropriate math(s) please explain how the proof works?
  • by Cryogenes (324121) on Thursday June 10, 2004 @04:45AM (#9385270)
    but unfortunately it was censored by the Slashdot lameness filter.

In seeking the unattainable, simplicity only gets in the way. -- Epigrams in Programming, ACM SIGPLAN Sept. 1982

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