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Riemann Hypothesis Proved? 454

Posted by Hemos
from the cracking-the-problems dept.
Theodore Logan writes "Has the Riemann Hypothesis finally been proved? The proof is a couple of months old, and to the best of my knowledge a Swedish newspaper is the only one to take up the story yet, so there is certainly a possibility that this is a hoax, or a less than watertight proof. But if it turns out to be the real thing, it will, apart from winning the authors eternal fame and glory for finding the holy Grail of modern math, provide them with a cool $1 million as they claim the first Millennium Prize." We had a story a while back about this as well.
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Riemann Hypothesis Proved?

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  • by YellowSnow (569705) on Monday March 03, 2003 @11:17AM (#5424070)
    First Proof?
  • um... (Score:4, Funny)

    by Anonymous Coward on Monday March 03, 2003 @11:19AM (#5424086)

    "A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective eigenfunctions \psi_s (t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z (s') = Z (1-s'), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to \beta - s (\beta a real), and s' goes to 1 - s', which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s' of the \zeta. It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros located at the points s_n = 1/2 + i \lambda_n in the complex plane."

    What the fuck?

    • They don't call it "news for nerds" for nothing...
    • Re:um... (Score:5, Informative)

      by shayborg (650364) on Monday March 03, 2003 @11:35AM (#5424213)
      This [dgl.com] provides a pretty good layperson's overview of the problem, as well as why it's relevant.

      -- shayborg
      • by thatguywhoiam (524290) on Monday March 03, 2003 @12:02PM (#5424400)
        This [dgl.com] provides a pretty good layperson's overview of the problem, as well as why it's relevant.

        While the linked site does provide a layman's interpretation of the topic, when you first click to that page you are presented with:

        Two Plus Two Equals Four

        Thought we had all been trolled... :)

      • After reading the article [dgl.com], the solution looks pretty obvious.

        The Riemann hypothesis asserts that all interesting solutions of the equation z(s) = 0 lie on a straight line.

        Well, graphing s versus z(s), if z(s) is always zero, then obviously all values of s solving that equation lie on the s axis, which is a straight line. QED.

        Sheesh. That wasn't so hard.
    • by MarvinMouse (323641) on Monday March 03, 2003 @11:36AM (#5424227) Homepage Journal
      A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented.

      We are going to show you beyond a shadow of a doubt that the non-trivial zeros of the zeta-function are of the form 1/2 +- i*theta_n.

      It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective eigenfunctions \psi_s (t), parameterized by continuous real indexes k and l.

      To do this, we are going to use the operators D^{(k,1)} and their respective vectors \psi_s (t), such that using D^{(k,1)} on \psi_s (t) will produce k*(\psi_s (t)), where k is some non-zero constant. Unfortunately though, we have to show a way to product all of these operators. So the "construction of" the operators will be contained within the proof.

      Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function.

      These \psi_s (t) vectors are also all at "right-angles" to eachother. So their cross products = 0.

      Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z (s') = Z (1-s'), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to \beta - s (\beta a real), and s' goes to 1 - s', which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s' of the \zeta.

      Z(s') = Z(1-s') is true. Thus, we can show that there is a connection between the follwing symmetries:
      t goes to 1/t,
      s goes to \beta -s (where beta is a real number),
      and s' goes to 1 - s'

      In Q.M. we can show then a correspondence between one of these orthogonal states to a unique vacuum state (from Quantum Mechanics), and thus a solution of the zeta function.

      It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros located at the points s_n = 1/2 + i \lambda_n in the complex plane.

      From these neat little tricks, we can show that the Riemann Hypothesis must be true, because these things are true.
      • whew. Well THAT explains THAT then!

        Honestly, you basically just translated that gobbledeegook from Latin to French for me. I still don't really understand what it all means, but I shall now do what I have done in the past for articles related to extremely complex mathematical hypothesis (hypothesese?)... I'll just nod my head, tell myself "Sure! But of course!" and move on to look for more "+5 Funny" comments.

        Then maybe get back to work too. :)
      • by Some Bitch (645438) on Monday March 03, 2003 @12:12PM (#5424461)
        We are going to show you beyond a shadow of a doubt that the non-trivial zeros of the zeta-function are of the form 1/2 +- i*theta_n.

        I was with you right up to the point where you started typing.

        To do this, we are going to use the operators D^{(k,1)} and their respective vectors \psi_s (t), such that using D^{(k,1)} on \psi_s (t) will produce k*(\psi_s (t)), where k is some non-zero constant. Unfortunately though, we have to show a way to product all of these operators. So the "construction of" the operators will be contained within the proof.

        ERROR: STACK OVERFLOW! SYSTEM POWERING DOWN...

      • by Axiom_1 (564687) on Monday March 03, 2003 @12:49PM (#5424735)

        Construction Worker Joe: I think the non-trivial zeros of the zeta-function are of the form 1/2 +- i*theta_n

        Construction Worker Larry: I agree. It's clear when you consider the operators D^{(k,1)} and their respective vectors \psi_s (t)

        Construction Worker Joe: Of course, so long as using D^{(k,1)} on \psi_s (t) will produce k*(\psi_s (t))

        Construction Worker Larry: Yeah. Joe, you'd better make sure the Eigenvectors of those two boards you're nailing together have a dot product of zero. The last time the boards weren't orthogonal, the boss had a fit!

        Construction Worker Joe: Yeah, whatever. Hey, check out that girl's hyperbolic curves!

        • by jlcooke (50413) on Monday March 03, 2003 @02:34PM (#5425491) Homepage
          Construction Boss: cross product you moron! The dot product of zero between the nail and the board means you don't have a board, or you don't have a nail in which case I'll have to spring for workman's comp...but this is canada...so go for it!
          • Construction worker Joe: Yes sir, of course sir.
            Construction Boss wanders off...
            Construction worker Larry: Stupid pointy haired fuck, doesn't know that the dot product of two perpendicular vectors is 0.
            Construction worker Joe: Yeah, hell, the fucking cross product isn't even a scalar!
            Construction worker Larry: (in a serious tone) Now be fair, he could have meant (snicker) the zero vector. (snicker)
            Construction worker Joe: (laughing) I suppose if I was sliding the nail along the stud.
            the scene devolves into derisive snickering.
        • Foreman: I didn't pay you to discuss the Riemann Hypothesis, I paid you to hammer some nails into some wood! Get back to work or you'll be begging for work at the local university again.
      • Frink: Here is an ordinary square --

        Wiggum: Whoa, whoa, slow down, egghead.

        Frink: -- but suppose we extend the square beyond the two dimensions of our universe, along the hypothetical Z axis, there.

        Everyone: (gasps)

        Frink: This forms a three-dimensional object known as a "cube", or a "Frinkahedron" in honor of its discoverer.

        Homer: Help me! Are you helping me, or are you going on and on?

        Frink: Oh, right. And, of course, within, we find the doomed individual.
    • Oh My GOD, if I hear one more person use the word "orthogonal"...

      EVERYONE is using that damn word now to explain something. Buisness people, marketing people, GET OVER IT!!!!
    • Re:um... (Score:4, Insightful)

      by Asprin (545477) <gsarnoldNO@SPAMyahoo.com> on Monday March 03, 2003 @01:37PM (#5425052) Homepage Journal

      Please, I get enough of that kind of attitude on TV,
      in movies,
      at work,
      at home,
      at church,
      at the bus stop,
      in stores,
      at the DMV,
      in restaurants,
      under my bed
      and standing next to ANYONE who has vowels in their names.

      /. is, in fact, the only place since grad school where this sort of news topic discussion is **possible**, let alone normal, so I would appreciate it if you would NOT remind me how much more life is like Empty-V than The Discovery Channel, mmmkay?

      Thank you.

  • by siliconwafer (446697) on Monday March 03, 2003 @11:28AM (#5424154)
    Finally, the mathematics we need to generate the 1.21 jigawatts to go ...

  • Wow ... (Score:3, Informative)

    by shayborg (650364) on Monday March 03, 2003 @11:29AM (#5424156)
    IIRC this has been something of a holy grail for mathematicians for years. When John Nash tried to prove it in his prime, everyone laughed at him, not believing that even someone as brilliant as Nash could solve the infamous problem. If this isn't a fake, it's a pretty amazing achievement.

    -- shayborg
  • No one noticed this? (Score:5, Interesting)

    by epong (561351) on Monday March 03, 2003 @11:30AM (#5424167) Homepage
    This was posted in August, and no one noticed until now? I check the arXiv every single for new articles. But I never saw this. The reason: it was posted in general mathematics and high energy physics theory, not the number theory section where it would belong.
    The arXiv will post nearly anything that resembles a mathematical paper-they don't do any refereeing. However, they apparently use the "general mathematics" section for papers that seem crankish like this one. And the fact that it took more than six months for this proof to make the news is proof that absolutely no one reads that section.

    I haven't looked at the proof yet, but I'm worried that it will be at best a "physicist's proof"-a series of claims deduced by using some sort of physical reasoning that is not mathematically rigorous, since it seems to have been written by physicists, and is in the physics section.
    • by You're All Wrong (573825) on Monday March 03, 2003 @01:05PM (#5424856)
      Your worries are well founded. Mathematicians have seen it, and have dismissed it as being very sloppily, and _non-mathematically_ written. Noone I know (mathematicians, that is) have bothered to read beyond the point where they throw their hands up in the air and cry "this is nothing but sloppy handwaving".

      For example, in the _very first_ equation, he introduces an "operator", and conveniently forgets to mention what space this operator is supposed toact on. A Banach space? A Hilbert Space? We should not have to _guess_ what algebraic structure they're using.

      YAW.
  • Translation (Score:5, Funny)

    by Quixote (154172) on Monday March 03, 2003 @11:31AM (#5424176) Homepage Journal
    Here's a "translation" from InterTran [tranexp.com]. Be gentle.


    Classic matteproblem able have got a solution
    One of mathematics superb olösta problem, the so call Riemannhypotesen, able now have got a solution, 144 year after that that Bernhard Riemann publish sina mind if a special equation, as in its luck is förknippad with primtalen.

    If the really am exposing themselves that certificates am holding able they two problemlösarna so småningom give a receipt out one million dollar in reward.
    Year 1900 was gathering the world most outstanding mathematics to a conference in Paris. Wonder that present David Hilbert, the terms rankings mathematical think, 23 problem as sedan arrived that affect mathematics wonder heal 1900- digits, and than today. One of these problem each Riemannhypotesen, and defiance superb efforts has it stay olöst. IN November 2002 publish yet Carlos Castro from Clark Atlanta University, Atlanta, America, and Jorge Mahecha from University perceive Antioquia, Centre queue, Colombia a one proposal to solution.

    The reward on one million dollar as exhibitor of American Clay Mathematics Institute sheep they yet nots out traitor one year behind publication. This for that second mathematics bark poll term that police accomplishment and watch if the really is accurate.
    - About this is truly is the a grand sensation. This is a creature of problem as am claiming great effort entrance a eventual solution able verify. Tusentals mathematics the world over will pounce this and inspect certificates with a magnifier, says Andes Karlqvist, mathematics, professor in data and manager for Polarforskningssekretariatet.

    He am declaring that certain of Hilberts problem rather is problemområden than separate problem. A bit had also word if under these term as gone. If now Riemannhypotesen is absolved so is tens of they 23 problems absolved, seven is olösta, five is part absolved and one is nots inferior current.
    Under Andes Karlqvist each Hilbert really grand within sits precinct, with him was concluding a epok. He each the lastly as had survey over heal the mathematical science.

    Mathematics have the latest decade deployed very quickly, and the cheers article one aid as it olds 1900- digits mathematics nots be able anticipate datorn. Day all prompt and major datorer able manipulate huge amount speech and on short term make computations as formerly each impossible for a mans although he/ she was working a good deal currency with sina figure.

    With datorernas help had certain problem absolved, as fyrfärgsproblemet. The says that the nots ring up to more than four various colours for that färglägga a maps so that nots area with a common limit had same colour. One datorprogram had systematic gone through all conceivable alternative.

    Andes Karlqvist deem yet that the find one philosophy dilemma with this: inquiring is if husband bark accept evidence in form of one datorprogram. He am believing that wes now am standing before one kulturskifte within mathematics. Wonder the next decade am arriving the that evolve radically, and the because they all efficient datorerna.

    Bengal Jonsson

    • *laughs* This is just great fun to read ;)

      For everybody else, I speak Swedish as well, and it so sounds like a Swede who can't speak proper English, and adding Swedish words in the middle of everything.

      No offense to the original poster, and bork bork bork everyone.
      • Gotta love Swingligh - I work for the US arm of a Swedish company, and some of the memos that come down are hilarious. But damn, they never send us meatballs!
    • From a Swede (Score:4, Informative)

      by Anonymous Coward on Monday March 03, 2003 @11:37AM (#5424233)
      Here's a slightly better one [nada.kth.se].

      Slashdot them to hell. It's my university, they can take it.
    • Human Translation (Score:5, Informative)

      by erl (7024) on Monday March 03, 2003 @12:02PM (#5424398)
      That computer translation is hilarious!

      Here's a human translation:

      Classical Math Problem May Be Solved

      One of the great unsolved problems of mathematics, the so called Riemann Hypothesis, may have a solution 144 years after Bernhard Riemann published his idea of a special equation, related to prime numbers.

      If the proof does turn out to be correct, the two problem solvers may be eventually be able to collect a one million dollar reward.

      In the year 1900, the world's leading mathematicians gathered for a conference in Paris. During the conference, David Hilbert, the leading mind of mathematics at the time, presented 23 problems which would affect mathematics for the ensuing century, and yet today. One of these problems was the Riemann hypothesis, and despite great effort it has remained unsolved. However, in November of 2002, Carlos Castro of Clark Atlanta University, Atlanta, USA, and Jorge Mahecha of the University of Antioquia, Medellin, Colombia published a proposed solution.

      They won't be able to collect the one million dollar reward offered by the american Clay Mathematics Institute until one year after publication. This is to allow other mathematicians time to check the result, and verify its correctness.

      - If this is true, then it is a sensational. This is a class of problems which requires much work before a possible solution can be confirmed. Thousands of mathematicians all over the world will cast themselves at this, and examine the evidence in minute detail, says Anders Karlqvist, mathematician, professor of informatics and head of the Polar Research Secretariat.

      He explains that certain of Hilberts problems are problem areas rather than individual problems. Some have also been reformulated. If the Riemann Hypothesis is solved, then ten of the 23 problems are solved, seven are unsolved, five are partially solved, and one is no longer relevant.

      According to Anders Karlqvist, Hilbert was a great man within his field, and a great period ended with him. He was the last man to have an overview of all the fields of mathematics.

      Mathematics have developed very rapidly during the latest decades, thanks to an aid that the mathematicians of the early 20th century couldn't predict - the computer. The ever faster and larger computers of today can handle vast quantities of numbers and quickly make calculations that used to be impossible for a person even if he or she spent an entire lifetime.

      With the help of computers, certain problems have been solved, such as the four color problem. It says that at most four different colors are neccessary to colour a map, so that areas with a common border don't have the same color. A computer program has systematically gone through all possible combinations.

      Anders Karlqvist thinks that this involves a philosophical dilemma: should proofs in the form of computer programs be accepted? He believes that we stand before a culture shift within mathematics. During the coming decade mathematics will develop radically due to the increasingly efficient computers.

      Bengt Jonsson

      • They won't be able to collect the one million dollar reward offered by the american Clay Mathematics Institute until one year after publication. This is to allow other mathematicians time to check the result, and verify its correctness.

        And, what if the standard of refutation? Is it enough to claim "oh, this proof is all just handwaving", or "this proof is worthless, it uses a physicists approach", or does any detractor need to precisely pinpoint where the error is "on page 13, where they get from equation 63 to 64, they effectively multiply both sides with zero"?

        Indeed, it appears that most mathematicians don't take the proof seriously, which also means that nobody is taking the time to check it through... Thus ironically, our hoaxters may be able to collect... which will turn out very embarrassing to the contest board, if 5 years from now some bored math student goes through it line by line, trying out all possible interpretations, and does find the error(s)...

        • "And, what if the standard of refutation? Is it enough to claim "oh, this proof is all just handwaving", or "this proof is worthless, it uses a physicists approach", or does any detractor need to precisely pinpoint where the error is "on page 13, where they get from equation 63 to 64, they effectively multiply both sides with zero"?"

          There is a saying sometimes employed by cruel mathematicians to describe illucid 'proofs': "This isn't right. This isn't even wrong."

          Having said that, it surely would be nice to see 'exactly where it goes wrong'. For simple arguments a standard "the first error is on page t, line s" will suffice. More convenient is to find a claim to which an explicit counterexample can be constructed, that is, to show that the proof (if valid) leads to a contradiction - of course this works best when the proof is not reductio ad absurdum :-P

          Finally, we should always remember that the burden of "proof" is on the "prover" - it's not anyone else's responsibility if the paper isn't even written in what we'd call mathematics. (Naturally it has no chance of being published in a print journal if it's in that state, which is a necessary condition for the awarding of the prize.)

          (The rest is a slightly tangential discussion of two common problems arising from extremely imprecise methods, aka "handwaving".)

          Here's a gratuitous example:

          Prop. There are more real numbers than integers.
          "Proof." Consider the interval [n,n+1) for arbitrary integer n. In this interval there is one integer, but a slew of reals, eg, n+1/2, n+pi/4, and so forth. So, there are more reals than integers.

          Of course this proposition is true but the proof is nonsense, since we can "derive" by mimickery

          Prop. There are more rational numbers than integers.

          which is just false.

          This is a pretty simplistic example and doubtless the author of the purported proof of RH is using much more sophisticated handwaving, for which I'll produce another analogy:

          Prop. 2 has no proper factorisation.
          "Proof." The only integers dividing 2 are 1, -1, 2, -2, the former two being units and the latter two the number and an associate.

          This proposition falls into the "not even wrong" category. Why? -- because to write down the word "factorisation" begs the question: in what ring (ie, algebraic context)? Absolutely, 2 is irreducible in the integers. But not in the Gaussian integers:

          Prop. 2 is not irreducible in the Gaussian integers.
          Proof. We see easily that 2 = (1+i)(1-i). Each factor of the RHS has norm 2, so neither is a unit, and in fact 1+i is a Gaussian prime, and 1-i = (-i)(1+i) is also a Gaussian prime.

          The error in the first proposition about 2 was that it made no reference to the context of the discussion. This is the sort of handwaving that occurs in the "proof" of RH. We can see it really makes a difference - in one context the prop. was true, and in another false; sometimes even to utter the word "factorisation" is to already condemn oneself to the "not even wrong" bin.
          • Prop. 2 has no proper factorisation.

            This proposition falls into the "not even wrong" category. Why? -- because to write down the word "factorisation" begs the question: in what ring (ie, algebraic context)? Absolutely, 2 is irreducible in the integers. But not in the Gaussian integers:

            Context is everything. And in the absence of meaningful context, assume the most simple meaning of factorisation is intended, i.e. decomposition into plain vanilla positive real integers.

            Not specifying context is by itself not necessarily an error. It's just sloppy writing, and makes the proof harder to read (the reader has to figure out what exactly is meant). Not specifying context only becomes an error if you start mixing to contradictory meanings. For example, if in your "factorization of 2 problem", you start two lines of reasoning, one in which you limit yourself to integers, and one in which you allow complex numbers.

            A smart reader (and who also has lots of time on his hands...) may check out the proof by trying out the various possible contexts. Either he finds one context where the whole proof makes sense, or he does indeed find an incorrect mixing ("on page 10, the authors work in the algebraic context of simple integers, while on page 15 they work in the context of Gaussian integers"). Of course, the problem here is that the thing is so sloppily written, with so many underspecified contexts that nobody is really willing to invest any time debugging it...

    • by PurpleBob (63566) on Monday March 03, 2003 @12:56PM (#5424786)
      its luck is förknippad with primtalen.

      I suppose if I were about to win a million dollars, my luck would be förknippad with primtalen, too.
  • You can help (Score:5, Interesting)

    by Slightly Askew (638918) on Monday March 03, 2003 @11:33AM (#5424187) Journal
    In case you're tired of looking for UFOs with SETI, you can use your spare CPU cycles to help prove/disprove this hypothesis here [zetagrid.net].
    • Re:You can help (Score:3, Insightful)

      by turgid (580780)
      You can't find a proof simply by number-crunching. Looking for SETI is a better use of processor cycles, until someone writes a program that can do abstract mathematics with the insight and ingenuity of a human being or better.
      • Re:You can help (Score:3, Insightful)

        by Alranor (472986)
        Really?

        You might want to mention that to the people who finally proved the 4 colour conjecture a few years back then.

        And anyway, even if you couldn't find a proof of this theorem through pure number crunching, you may be able to find a counter-example, which would be equally interesting.
        • Compared to the Riemann Hypothesis, it looks like your 4 Colour Conjecture is mathematically very trivial in that it could be proved by number-crunching. I reiterate when I say that where there are possibly an infinite number of soultions, merely numbercrunching will not give you a positive proof.
        • Re:You can help (Score:3, Informative)

          by saforrest (184929)
          The four-colour theorem was a graph-theoretic problem with a finite number of cases. This is not.

          Sure you might find a counterexample instead of a proof. But all the OP said was that you wouldn't find a proof.
      • Why not? One pet idea of mine is to make a distributed proof-finder for Fermat's Last (yeah, I know it's proven, but I want the ``elegant'' proof). Simply combine a combanitoric proof-generator with a proof-checker, and you've got a program which finds proof simply by number-crunching. Of course, I could argue that all the human brain is doing is number-crunching anyways...

        Of course, this isn't how the ZetaGrid people are doing it.
    • Seriously, not to be critical or anything, but I'm just curious:

      Why would you spend your spare CPU cycles on something like this? Why not put them more towards protein folding [stanford.edu] or an AIDS cure [fightaidsathome.org] or even evolutionary research [evolutiona...search.net]... something that would/might benefit humanity? Or is finding a proof/disproof to this hypothesis going to benefit us somehow?
  • by Anonymous Coward on Monday March 03, 2003 @11:34AM (#5424196)
    ...trivial and left to the reader. Sorry ;-).
  • by kip3f (1210) on Monday March 03, 2003 @11:34AM (#5424202) Homepage
    This item is a submission to a mathematics journal, it has not yet been accepted for publication. The authors claim to have proven the RH by a new method, and the editors of the journal are currently reviewing the article for errors. If they find no major errors, and the minor errors are fixed, then the article will appear in the journal. Then a much larger group of mathematicians will review the article. If, after all this time, the article hasn't been disproved, then it can be accepted as valid.
  • by levell (538346) on Monday March 03, 2003 @11:35AM (#5424215) Homepage
    I think that the proof of the Poincare Conjecture [arxiv.org]. Will get the first Millenium Prize (by a few days!) if it is correct (talked about here [slashdot.org] but hey if this gets the second then blimey! Two already!?! Could be a good decade for mathematics!
  • by arvindn (542080) on Monday March 03, 2003 @11:37AM (#5424230) Homepage Journal
    Definition of the zeta function: There is something called Riemann's zeta function: it is a function of a single complex variable. It is defined as zeta(z) = 1^(-z)+2^(-z)+3^(-z)+... (ad infinitum) You can easily see that zeta(z), as defined, converges if and only if Re(z)>1 (real part of z). However, the function is defined for all complex z using something called the analytic continuation: basically there is a unique way to extend zeta(z) for re(z) < 1 in such a way that derivatives of all orders exist at all points.

    The hypothesis states that all (nontrivial) zeroes of the zeta function occur on the line Re(z) = 1/2.

    If proved, it has immense implications in many areas of pure and applied mathematics. For instance, in number theory: it would say a lot about the distribution of prime numbers.

    The stature of the problem can be seen from the fact that it was one of the 23 problems which would shape the mathematical progress of the 20th century that David Hilbert drew up in his lecture at the 1900 Paris congress of mathematicians.

  • by siliconwafer (446697) on Monday March 03, 2003 @11:37AM (#5424236)
    Re:Could you get a bit more arrogant please? (Score:5, Informative)
    by njj (133128) on Tuesday July 02, @12:05PM (#3808279)
    (http://www.csv.warwick.ac.uk/~marem/)
    If you can't explain something in ordinary words to a layman, then you really don't understand it.

    I'm about halfway through writing up my PhD thesis on some applications of homological algebra to knot theory and low-dimensional geometric topology (provisional title liber rerum dementiae, but it'll probably end up being called something more mathematically appropriate).

    In principle, yes, I could explain the details of my research to a suitably motivated layman. But I suspect it would take rather a long time.

    You see, and this really isn't meant to sound arrogant, supercilious, or dismissive, university-level mathematics is pretty damned difficult, and the details of most cutting-edge research really doesn't make sense until you've spent several years learning the background (the mindset, the language, the fundamental concepts).

    My current area of research is essentially the applications of homological algebra to knot theory and low-dimensional geometric topology. To explain this to a non-mathematician, I'd first have to teach them a lot of background stuff (group theory, a bit of stuff about rings and modules, point set topology, basic algebraic topology (the fundamental group, (co)homology theory), some geometric topology (basic course in knot theory, some stuff about 3-manifolds), a bit of category theory, and some homological algebra (broad overview of the (co)homology theory of groups and algebras)).

    It's taken me nearly nine years (3-year BA, 1-year MSc specialising in topology and knot theory, plus nearly five years doing a (part-time) PhD) to get to this point myself. If I were a bit cleverer (or didn't have a `proper' job as well) I might have been able to shave a couple of years off that.

    My friend Steve has a physics degree. I managed, in ten minutes one evening, with much handwaving, to give him some idea of what my thesis is all about. It helped that he knew what a group was already though. But for me to explain it fully to him would probably necessitate him doing at least one mathematics degree first. And that's not really something I'd wish on one of my friends :)

    Now this really isn't meant in an arrogant way, and I hope you won't read it like that, but Euclid was right: There is no royal road to geometry.

    I can have a go at explaining the Riemann hypothesis, though. To fully understand what it's about and why it's so damned difficult you'll need to do an advanced course in complex analysis (which isn't my field either).

    A complex number is a sort of two-dimensional number, which you can regard as a point in a plane (the `complex plane' or `argand diagram'). You add them together coordinate-wise, and you multiply them together in a weird manner which involves something which behaves like a `square root of -1' (engineers also like to think of it as a sort of 90-degree phase-shift operator, I'm told).

    There's a particular function (`Riemann's zeta function') defined on the complex plane (it takes one complex number as input and returns one complex number). For some complex numbers (`the zeros of the function'), the value of this function is zero.

    The `trivial' zeros occur at the points -2, -4, -6, ... on the horizontal axis.

    The `non-trivial' zeros (that is, all the other points for which zeta is zero) all seem to occur on the line parallel to the vertical axis that intersects the horizontal axis at +0.5. Indeed, nobody's ever found one which doesn't.

    The Riemann Hypothesis is that *all* the non-trivial zeros lie on this line. It's known to be true for the first (large number which temporarily escapes me), but it turns out to be phenomenally difficult to prove that it's true in every case.

    Now that's the basic idea, but it doesn't (and I can't - it's not my field) explain *why* it's so difficult that some of the greatest minds (Hardy, Littlewood, Ramanujan, etc) of the past 150 years have failed to prove it, and why the Clay institute are willing to pay a million dollars to someone who can.

    - nicholas (we don't just sit around doing big sums, you know :)
  • "After Jim proved the hypotheses, the hypothesis was proven."
    • by kfg (145172)
      Or it was proved. Like it or not, either is both common and acceptable, with proven actually being a modern upstart or "incorrect" derivation by analogy with other forms.

      "Usage Note: Prove has two past participles: proved and proven. Proved is the older form. Proven is a variant. The Middle English spellings of prove included preven, a form that died out in England but survived in Scotland, and the past participle proven, a form that probably rose by analogy with verbs like weave, woven and cleave, cloven. Proven was originally used in Scottish legal contexts, such as The jury ruled that the charges were not proven. In the 20th century, proven has made inroads into the territory once dominated by proved, so that now the two forms compete on equal footing as participles. However, when used as an adjective before a noun, proven is now the more common word: a proven talent."

      Go figure.

      KFG
  • Heh (Score:4, Interesting)

    by pclminion (145572) on Monday March 03, 2003 @11:39AM (#5424249)
    I guess John Nash can rest easy, now!

    The Riemann hypothesis isn't exactly the most practical of problems, but many people have spent decades working on it (and some have gone insane). It's good that it is finally put to rest.

  • by King Babar (19862) on Monday March 03, 2003 @11:45AM (#5424299) Homepage

    I know the editors of this site mean well, but what we have here is a link to a site that defines the Riemann Hypothesis in very abstract terms, a link to a LANL preprint from two completely unknown researchers deposited there in November 2002, and a link to an obscure Swedish newspaper from almost two weeks ago, and no other supporting material. So my BS meter is running at 5.

    The odds that "this is the one!" given that pedigree would seem to be really tiny. But the clincher for me is the following web page dedicated to would-be proofs of the Riemnann Hypothesis [ex.ac.uk] whose important text is (and I quote):

    If you are a university mathematics lecturer who teaches analytic number theory, you might want to consider setting your students the task of deconstructing the more serious of these. They may otherwise never be given any serious attention, which would be a shame.

    And the Castro and Mahecha preprint (and another grandiosely titled preprint by Mahecha) is linked to from there. Now my BS meter is running at about 9. So now I check for messages abou this at deja.com in the sci.math group. [google.com] Read the thread yourself; it's pretty entertaining.

    So, with my BS meter running at 11, the work having been submitted for coming up on 6 months, and no indication whatsoever that this is real, I suggest it is false.

    And I also suggest that Slashdot might wish to consider contacting a real mathematician to filter their potential stories on mathematics, since I can't tell you the last time one of these "is X finally proven?" stories has panned out.

    • by Theodore Logan (139352) on Monday March 03, 2003 @12:07PM (#5424427)
      Yeah, I found both of those link about half an hour after I submitted the article. Still, I'd like to add some corrections.

      1. SvD isn't an "obscure" Swedish newspaper. It's the biggest, counting readers in if not millions so at least hundreds of thousands. They seldom print bogus.

      2. That the proof hasn't been verified yet doesn't mean it can't be correct.

      3. The sci.math discussion doesn't really say anything about the validity of the proof, only that, as you say, the paper seems to not have been proof-read very well, etc.

      But, I agree that in essence you have reasonable complaints. BS meter at 11 is quite high, though. Mine is at about 5. 11 is reserved for make-money-fast schemes and herbal viagra.

      Oh, and one more thing. An AI posted a translation of the article [slashdot.org] that seems to have gone largely unnoticed. (Just in case there is someone in here who isn't fluent in Swedish.)
      • by King Babar (19862) on Monday March 03, 2003 @12:25PM (#5424571) Homepage
        1. SvD isn't an "obscure" Swedish newspaper. It's the biggest, counting readers in if not millions so at least hundreds of thousands.

        While I have no doubt that SvD is of singular value to almost everybody (especially for its largest entries on the diagonal), the fact is that it comes from Sweden makes it orthogonal to our concerns. (Sorry about that...)

        Here in the US, you have to understand that unless you share a border with Iraq, we just don't have time to be interested in you these days. :-)

        2. That the proof hasn't been verified yet doesn't mean it can't be correct.

        True enough, but see below, and the fact that if *I* had a proof of the Riemann Hypothesis, I probably wouldn't submit it to the high energy physics and "general math" sections of xarchiv.

        3. The sci.math discussion doesn't really say anything about the validity of the proof, only that, as you say, the paper seems to not have been proof-read very well, etc.

        Actually, the discussion basically says that nobody could read the thing and that it was chock full of typos. Once again, if I had a proof of the Riemann Hypothesis, I would probably make very certain that it was free of such interest-busting material. A mathematician can feel free to correct me, but I am assuming that while raw papers (especially from non-native speakers of the language the paper is submitted in) can be tough to get through, you usually *do* detect the high quality of the real work at some point pretty early on, and if you don't, you assume the worst.

        In the thread I referenced, a physicist chimed in with the observation that one of the co-authors (Castro) was not taken seriously in his own field of physics, which makes it even less plausible that the math in this paper would be new and inspiriational.

        But, I agree that in essence you have reasonable complaints. BS meter at 11 is quite high, though. Mine is at about 5. 11 is reserved for make-money-fast schemes and herbal viagra.

        Unfortunately, the Clay Math Prize has kind of made proving the Riemann Hypothesis a "make-money-fast" scheme. :-)

        Or to put it another way, here's the plan of attack I believe the authors had:

        1. Write an incomprehensible paper on the Riemann Hypothesis and sling it up onto the xarchiv.
        2. ???
        3. Profit from the Clay Math Prize!
        • by Theodore Logan (139352) on Monday March 03, 2003 @01:20PM (#5424940)
          While I have no doubt that SvD is of singular value to almost everybody (especially for its largest entries on the diagonal), the fact is that it comes from Sweden makes it orthogonal to our concerns. (Sorry about that...)

          I don't understand what you mean by this, or if you are even being serious. If a newspaper is respected, generally trustworthy and read by a large amount of people on a daily basis, where is happens to be located should of course be of no relevance.

          If you mean to say that the reason that the rest of the media isn't catching on is that the only story so far has been in a Swedish newspaper, I don't object. But if you're saying that being run in a non-US newspaper makes the story less likely to be true, I think you are a little confused.

          True enough, but see below, and the fact that if *I* had a proof of the Riemann Hypothesis, I probably wouldn't submit it to the high energy physics and "general math" sections of xarchiv.

          Recently, as in the last couple of years, the most promising contributions to RH related stuff has come from high energy physics, and many people, both in math and in physics, believe that this is the approach that will eventually bear fruit. If indeed a final proof would be more physics than number theory (perhaps the proof of the RH would only be a corrollary of some completely un-number theoretic line of reasoning) wouldn't it be reasonable to publish it in a physics journal? That it has implications for number theory isn't enough of a reason to publish it in a number theoretical journal, mostly because the readers of it would not be able to determine whether it was correct or not.

          In the thread I referenced, a physicist chimed in with the observation that one of the co-authors (Castro) was not taken seriously in his own field of physics, which makes it even less plausible that the math in this paper would be new and inspiriational.

          This is serious, agreed. I don't think I read that post.

          Unfortunately, the Clay Math Prize has kind of made proving the Riemann Hypothesis a "make-money-fast" scheme.

          Only for amateurs. But those have been trying to prove it, as well as Fermat's last theorem, Goldbach's conjecture, the twin prime conjecture etc. for a long time already. Serious researchers, on the other hand, very rarely put their reputation at stake if they don't believe they have something of real value. They know flaws will be detected, and they know that they would never win any prize with an unsound proof.

          But, like I said (in the write up, even), there is a clear possibility that this isn't the real thing. I only think you're overstating your case.
          • While I have no doubt that SvD is of singular value to almost everybody (especially for its largest entries on the diagonal), the fact is that it comes from Sweden makes it orthogonal to our concerns. (Sorry about that...)
            I don't understand what you mean by this, or if you are even being serious.

            I wasn't. It was a joke. A really bad joke. Sorry about that. :-)

            If a newspaper is respected, generally trustworthy and read by a large amount of people on a daily basis, where is happens to be located should of course be of no relevance.

            It isn't, except that it is. The problem I have here is with accepting "generally trustworthy" as a blanket statement that applies equally to things we know newspapers are usually pretty good at (politics, current issues, scandals, crimes, and the like) and areas where we (or at least I) do not have high confidence in their abilities. High-level mathematics clearly falls into the second category; I am not certain I would trust *any* newspaper account of this. Fort that matter, I can get specific here. The translation of the article you pointed us to includes this:

            One of these problems was the Riemann Hypothesis, and despite great efforts it has remained unsolved. But in November 2002 Carlos Castro from Clark Atlanta University, Atlanta, USA, and Jorge Mahecha from the University of Antioquia, Medellin, Columbia, offered a solution.

            Now, the problem here is that some random author or another submitting a paper that claims to solve an important math problem like this is really not newsworthy. It really does happen almost all of the time, and yet the writer of this story seems to be innocent of this fact. Indeed, if the author had asked for comment or advice from any mathematician about this particular attempt, I'm sure the answer would have been "we see five or six of these a year". Actually, the fact that this is a *good* newspaper makes the point even more strongly: despite their expertise, they really didn't know how to evaluate this as a news story. That's the problem.

            True enough, but see below, and the fact that if *I* had a proof of the Riemann Hypothesis, I probably wouldn't submit it to the high energy physics and "general math" sections of xarchiv.
            Recently, as in the last couple of years, the most promising contributions to RH related stuff has come from high energy physics, and many people, both in math and in physics, believe that this is the approach that will eventually bear fruit.

            OK, now that's interesting, and I did not in fact know that. And there is no question that new lines of attack can come from unusual places. I do feel that these new lines often offer themselves up in somewhat more propitious circumstances. So if a high energy physicicst had teamed up with an important number theorist to do the paper, and it was based on the kind of insight that is taken for granted in one community but not the other (e.g., "interesting; in our work we rewrite the integral *this* way and then prove these bounds...")

            Unfortunately, the Clay Math Prize has kind of made proving the Riemann Hypothesis a "make-money-fast" scheme.
            Only for amateurs. But those have been trying to prove it, as well as Fermat's last theorem, Goldbach's conjecture, the twin prime conjecture etc. for a long time already. Serious researchers, on the other hand, very rarely put their reputation at stake if they don't believe they have something of real value. They know flaws will be detected, and they know that they would never win any prize with an unsound proof.

            Well, the term "amateur" is always a relative thing. So it would be easy to see why a non-famous high energy physicist might spend a lot more time on something like RH after the prize money is publicized given that the chances of hitting on this or on any round of funding these days in physics might be the same lottery odds. :-( So a detectable number of physicists entered the field of cognitive neuroscience when the SSC went down, not necessarily for the big bucks, but to try and work on a challenging problem where there was some hope of funding. Results were mixed.

            Plus, I can report that a couple of definitely professional mathematicians I have met do (prviately) admit that the Clay Prize money has actually attracted their attention to the prize problems, if only just to take a brief whack at them and see if anything new falls into place.

            But, like I said (in the write up, even), there is a clear possibility that this isn't the real thing. I only think you're overstating your case.

            I have pretty high confidence that the current attempt has basically already joined all of the other failed attempts. There are so many of these, and this does share many disturbing similarities with them. Thanks for your comments, though; I really had no idea that high energy physics had any implications for RH until I read your post.

    • by arvindn (542080) on Monday March 03, 2003 @12:15PM (#5424482) Homepage Journal
      I used to be in a number theory mailing list, and every week or so the list would get a mail from some amateur who thought they had proved this or that Longstanding Conjecture (TM). Everybody's favorite was the Goldbach conjecture, with the twin prime conjecture coming in a close second. Usually anyone who with a single number theory course in college could spot the error in a 10-20 line "proof". Sometimes it would be somewhat longer MS-word document that weren't so obvious. Proofs written in TeX were rare, and would need a number theorist to point out the flaw.

      There must be hunderds of these "final proof of Riemann hypothesis" claims on the web. It is sad that "a Swedish newspaper is the only one to take up the story yet" doesn't inspire caution in the /. editors but urges them on to more recklessness.

    • That was in 1994. If you read the following link, you'll notice that the first time the proof was presented, it had a fatal flaw, and it wasn't corrected until over a year later. Even if this Riemann proof eventually pans out, it is almost certain that at this point, it has some serious flaws.

      http://www.missouri.edu/~cst398/fermat/contents/ th eorem.htm
    • BS meter (Score:3, Informative)

      by jbolden (176878)
      My BS meter is running even higher. The whole structure of the document uses a very primative notion of zeta functions and the most specific version of the Riemann hypthosis. In other words the version that has derailed hundreds of brilliant people. If a proof is going to be presented at this point its either going to be an indirect attack (like the proof of Fermat's last theorem based on Fry) or a conceptual breakthrough in another field. In neither case is the problem likely to be phrased in this 18th century style.

  • by MjDascombe (549226) on Monday March 03, 2003 @11:48AM (#5424306) Journal
    Can be found here [claymath.org]
  • ...tell us if this will make factoring large numbers easier? What are the implications for public key encryption?
  • The sci.math thread is archived at http://groups.google.com/groups?&threadm=3D749A9C. 1B651415%40hotmail.com . Comments appear luke-warm.

    Note that the paper was submitted to the "High Energy Physics" archive, not the "Mathematics" archive. The abstract has some physics jargon, too. What this means for the proof I cannot say.
  • Breaking Encryption? (Score:2, Interesting)

    by phorm (591458)
    If it becomes simple to factor the product of prime numbers, current digital encryption software will be worthless.

    How does this make encryption software worthless? Being able to unfactor the primes wouldn't seem to me like it would automagically be the solution to cracking an encryption key, etc. Even a program could unencrypt a document by guessing various keys etc through prime factorization (I'm assuming that is what this is about), how would it know which solution is right?
    • Because you multiply the primes and you get the number you where trying to factor?
    • by mark-t (151149)
      While I agree it wouldn't make encryption worthless, it would render all existing common encryption systems as such. Right now, their security is resting *entirely* on the magnitude of difficulty of the problem of factoring a large number that happens to be a product of exactly 2 primes. The premise being that if it is difficult enough for someone to decrypt, by the time they succeed, hopefully the information will be worthless. If prime factoring became easy enough (in terms of big O), it would be conceivable for anyone to decrypt materials that were supposed to be secure in a feasable time frame.

      Quantum encryption, IMO, is the answer - if you even *try* to decrypt it, you muck it up and both the sender and receiver know about it.

  • by Anonymous Coward on Monday March 03, 2003 @11:58AM (#5424373)
    Although the way it's formulated is fairly complex, it is intended to answer the question:

    How many prime numbers are there less than a given number.

    It doesn't take much thought to work out why that would be handy in say cryptography.

    But most complex maths starts for it's own sake. You build the tools in the knowledge that eventually someone is going to use them, and inevitably they always do.

    I read about advances in nano-technology all the time. What's the point, no-one's using them? But without them now we wouldn't have cool stuff in 20 years. Same goes for maths. I would have thought nerds of all people would get that point.
  • by Anonymous Coward on Monday March 03, 2003 @12:01PM (#5424397)
    "A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective eigenfunctions \psi_s (t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z (s') = Z (1-s'), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to \beta - s (\beta a real), and s' goes to 1 - s', which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s' of the \zeta. It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros located at the points s_n = 1/2 + i \lambda_n in the complex plane."

    That is the WORST pickup line I have ever heard.
  • *siiiigh* (Score:3, Offtopic)

    by Com2Kid (142006) <com2kidSPAMLESS@gmail.com> on Monday March 03, 2003 @12:05PM (#5424416) Homepage Journal
    It seems for every bit of knowledge I gain, I just learn how much more there is to know.

    Maybe in yet another two or three years I will be able to understand WTF they are talking about. . . .
  • But.... (Score:3, Funny)

    by Eu4ria (110578) on Monday March 03, 2003 @12:10PM (#5424449)
    will this help me get higher frame rates in Doom III? If not then it is useless ;)

    Eu4ria
  • I would think that rendering our current method of digital encryption (factoring very large prime numbers) absolutely useless would be worth a little more than a million bucks! i would be expecting something more along the lines of the NSA's budget for a year or something like that...but, hey, thats just me...
  • by Powercntrl (458442) on Monday March 03, 2003 @12:13PM (#5424467)
    Will it lead the way to the return of an integer to represent your karma?
  • Why its important.

    Almost all "good" encryption uses prime numbers. If someone can figure out how to factor prime numbers, or find a quick way to determine a prime number, then all the fancy encryption in the world wont help because someone can just crack it in real time.

    Right now public encryption works because it would take so long to break the encryption, even with say... 300k computers (distrubted.net) that when the encryption was broken (5-10 years later) the info would be old and it would not matter.

    If someone could figure out some way to factor primes (which is along the lines of riemans sums) then they could possibly break even our stronger prime based encryption, which would make a great many people have to go back to teh drawing board on encrypting their data.
  • So, this is an attempt at a "very hard" proof of a function that helps generate prime numbers, correct? It seems to me that the lack of a rock-solid proof wouldn't keep more pragmatic-minded crypto software developers from just "assuming" that RH can be used (1.5*10^9 examples aren't often wrong), and working from that assumption. I mean, the only tractable methods of generating them as it stands are probabalistic anyway, so the only way to go would be up. Does anyone know if any software uses the zeta function to generate/verify large primes?
  • by peter_gzowski (465076) on Monday March 03, 2003 @01:31PM (#5425008) Homepage
    so there is certainly a possibility that this is a hoax, or a less than watertight proof.

    I read this as waterproof tights for some reason... had visions of Batman in a wetsuit.
  • by N8F8 (4562) on Monday March 03, 2003 @02:03PM (#5425254)
    The theorem is explained here [ou.edu] with pictures.
  • by lingqi (577227) on Monday March 03, 2003 @08:17PM (#5428646) Journal
    from Mathworld [wolfram.com] and I quote:

    In 2000, Clay Mathematics Institute offered a $1 million prize for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line), does not earn the $1 million award.

    An example that operates on the exact opposite principle of awarding prizes is the recent battle between Kasprov and Deep Jr: He gets 500k regardless, and 300k extra if he wins, 200k extra if he loses, or 250k if draw (i think the last case took place).

    Talk about being stingy! I'd think that disproving the Riemann Hypothesis would be equally interesting as proving it - There are soooo many theorms out there that basically begins with "We assume that the Riemann Hypothesis to be true, and so forth so forth."

If Machiavelli were a hacker, he'd have worked for the CSSG. -- Phil Lapsley

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