Slashdot Mirror

← Back to Stories (view on slashdot.org)

A Step Towards Proving the Riemann Hypothesis

Posted by kdawson on from the prime-time dept.

arbitraryaardvark writes "A new mathematical object has been discovered by Bristol University student Ce Bian. The Riemann hypothesis, unproven since 1859, has to do with the distribution of primes and something called L-functions. Bian has demonstrated the first known third-degree transcendental L-function. This apparently opens up a new way to go about looking for proofs of the Riemann hypothesis. There is an unclaimed $1 million prize for a valid proof. We've discussed a couple of earlier attempts to claim the prize."

15 of 133 comments (clear)

  1. My own personal proof by explosivejared · · Score: 4, Funny

    Non-trivial zeroes of the zeta function are 1/2 because they naturally form as wholes, but as we all know a grue can't resist the tasty flesh of a non-trivial zero. I posit that the only way to prove the hypothesis is to kill a grue and vivisect it to search for the other half of the non-trivial zero. So until someone is brave enough to fight a grue and extract the flesh of the non-trivial zero, that million dollars is going unclaimed.

    --
    I got a catholic block.
  2. I have already solved this! by Anonymous Coward · · Score: 4, Funny

    ...But unfortunately I do not have
    enough room in the margin of this
    text area to display it properly.

    1. Re:I have already solved this! by popmaker · · Score: 5, Funny

      am on ship on way to england stop have solved riemann hypothesis stop will give details on return stop

    2. Re:I have already solved this! by SpecTheIntro · · Score: 4, Interesting

      You know, there's a lot of speculation about that. I suspect he did have a proof, but I'm skeptical that it was correct. There's no doubt that the man was brilliant but we've had people working on that question ever since Fermat died and no one has been able to produce a "simple, elegant" proof. (Fermat's own description, there.) But there's plenty of precedent for mathematicians making things inordinately complex before some young genius comes along and shows a magnificently simple way of achieving the same thing.

  3. The Riemann Hypothesis, by Robert Ludlum by HiggsBison · · Score: 5, Funny

    Cue the creepy, hushed voice-over:

    In a University in Lower Saxony, a mathematician had formulated a remarkable conjecture. Its effects would be felt worldwide.

    The Riemann Hypothesis, by Robert Ludlum. Now in paperback.

    --
    My other car is a 1984 Nark Avenger.
  4. smile and nod by esocid · · Score: 4, Funny

    (just smile and nod, smile and nod. they'll never know you have no idea what this means)

    --
    Absolute power corrupts absolutely. indymedia
    1. Re:smile and nod by piemcfly · · Score: 5, Funny

      I can vouch for the smile-nod method.

      I got through half a year of classes on game-theory in Korean with it. The professor never noticed I didn't speak a word Korean. Tests were in English, and the guy just kept asking ending in '...isn't that right, studentX?' or '...do you not agree, studentY?'.

      Smile and nod baby, smile and nod. Best followed by a short single chuckle, as if the intrinsical irony of reality does not elude you.

      By the end of the semester the guy actually seemed to like me.

  5. Re:wow... by invisiblerhino · · Score: 4, Informative

    Actually, the Riemann hypothesis is pretty important, given that a proof of it would tell us about the distribution of prime numbers, and prime numbers are the wheels which keep e-commerce turning (RSA anyone?) Also, concerning scientific results which sound like Robert Ludlum novels, my own personal favourite is the Born Approximation - the least popular in the Bourne series.

    --
    xterm -n 8
  6. What's really going on here by l2718 · · Score: 5, Informative

    Booker and Ce Bian constructed certain degree 3 L-functions, but it is best to think of their discovery as follows: there is a complicated 5-dimensional membrane known to mathematicians as "SL(3,Z)\SL(3,R)/SO(3)". This membrane has subtle number-theoretic symmetries, so that its modes of vibration encode number-theoretic information. These modes (and their vibrational frequencies) are being extensively studied, but they are very transcendental objects so they cannot be written down explicitely and must be computed numerically. While certain modes (and frequencies) were already known numerically (they can be constructed from vibrational modes of the 2-dimensional membrane "SL(2,Z)\SL(2,R)/SO(2)" via something known as the Gelbart-Jacuqet lift) we now have the the first numerical computation of "native" modes of the 5-dimensional membrane -- those that aren't related to lower-dimensional cases. To each such mode of vibration there is an associated "L-function (similar to the Riemann zeta function), and it is the L-functions that were constructed. In fact, verifying that the approximate L-functions that were found correspond to actual modes of vibration is not easy (in the 2-dimensional case there is important work of Booker with others about this).

    In short, this is an important advance in automorphic forms, but it is so technical that it doesn't belong on SlashDot.

    It is important to realize that while indeed there is a ("Generalized") Riemann Hypothesis associated to these L-functions, numerically computing them represents zero progress toward proving the Riemann hypothesis for these L-functions or the original Hypothesis for the Riemann zeta function. At most this will allow very approximately computing some of their zeros and thus a weak check on the GRH for these L-functions.

    1. Re:What's really going on here by kalirion · · Score: 5, Funny

      Wow, it's so clear now!

    2. Re:What's really going on here by bjorniac · · Score: 4, Interesting

      Somewhat, but the parallel conjecture that went all the way back to Euclid couldn't be proven, even though it seemed largely true. Eventually Riemannian geometry arose as something that broke this well established conjecture. Often, yes, it's useful to assume conjectures, but don't underestimate the value of a proof, or even the value of failed proofs.

    3. Re:What's really going on here by l2718 · · Score: 4, Informative
      Let's try:
      • The "membranes", "modes" and "frequencies" here are already a physical analogy. Number theorists study objects (``automorphic forms'' -- no matter why they are called this way) that live on some ``manifolds'' (no matter what that means, either). But to get some intuition you can replace ''manifold'' with ''taut membrane'' (like a drum) and ''automorphic form'' with ''normal mode'' a.k.a. basic ''standing wave'', as you call it. An important problem in mathematical physics is to find what are the possible frequencies of standing waves on a particular surface. The problem here is analogous.
      • To see a picture of the 2-dim membrane I was talking about, see here. Start by taking a half-infinite strip of width 1, and cut off a semi-circular bit at the bottom like in the picture (the strip extends infinitely far at the top. Next, glue the two infinite sides together so the strip becomes a cylinder. Finally (that's not in the picture) imagine that as you go further and further up the cylinder, its radius becomes smaller and smaller, so the real thing is a kind of infinite funnel.
      • To see what a standing wave on this membrane looks like, see here (this was computed numerically by Dennis Hejhal).
      • The "lift" that takes a standing wave on this space to a standing wave on the 5-dim space is really complicated (and is a very indirect construction). There just isn't a non-technical way to describe it.
      • However, we know what the "lift" does to the frequencies: if you start with a standing wave you found numerically, and approximately know its frequency, then you know there will be a lifted guy of a calculatable frequency on the 5-dim space. So the interesting problem is to find standing waves with frequencies which are different from the ones we already know about (because we have calculated a lot of standing waves on the 2-dim surface).
      • One symmetry this infinite funnel has is left-right reflection (it is apparent both in the picture of the strip and in the picture of the vibrational mode). The other symmetries are difficult to describe in a blog post. What's important is that the modes of vibration must respect the symmetries.
      • It is true that to each such ''standing wave'' (on the 2-dim surface, on the 5-dim space, and on others) there is an associated L-function. The Riemann Hypothesis for these L-function (the same formulation: all zeros are on the critical line) is called the "Generalized (or Grand) Riemann Hypothesis" or GRH.
      • It was possible to calculate a few zeros of the newly-found modes, and see that indeed they are where they are supposed to be. This gives some evidence for the GRH. Calculations like this can always falsify the GRH (by finding a zero off the line). However, these calculations don't represent any progress toward proving the GRH -- that was confusion on part of the person who submitted the story to slashdot.
      I hope this helps.
  7. Do you know what you're talking about? by l2718 · · Score: 5, Informative

    Actually, what the RH tells us about the distribution of prime numbers is be pretty useless regarding RSA. To get anywhere you need the Extended Riemann Hypothesis (covering Dirichlet L-functions) and even the full force of the "Generalized Riemann Hypothesis" (covering all automorphic L-functions) is not known to help with the really important problem here -- factoring.

  8. I forgot to credit Marginal Revolutions blog by arbitraryaardvark · · Score: 5, Informative

    Submitter here. Right after hitting submit, I realized I'd forgotten to link to marginal revolutions, an economics blog that pointed me to the story.
    http://www.marginalrevolution.com/marginalrevolution/2008/03/assorted-link-4.html
    http://www.marginalrevolution.com/

  9. You miss the point by oni · · Score: 4, Informative

    infiltrating hundreds of thousands of computers to work on the solution

    The solution isn't to be found through massive computing effort. They are looking for a proof, not a computation. They need creativity, not horsepower.