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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."

14 of 133 comments (clear)

  1. 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
  2. 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 MaWeiTao · · Score: 2, Informative

      I sincerely tried to follow all that, but it's so far over my head that it's in orbit around Jupiter.

    2. 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.
    3. Re:What's really going on here by onemorechip · · Score: 3, Informative

      Paul Cohen. It was Paul Cohen.

      And he didn't solve the continuum hypothesis. He showed that you cannot prove CH from the ZF axioms. Gödel had previously show that you cannot *disprove* CH from ZF (unless ZF is inconsistent). Together these results show that CH is independent of ZF.

      So CH is still an unresolved problem today. As far as anyone knows, either CH or its negation can be taken as a separate axiom of itself, which leaves it an open question.

      --
      But, I wanted socialized health insurance!
  3. Please STOP reading Science Daily! by killmofasta · · Score: 1, Informative

    Science Daily is NOT science, nor Daily.
    The Holy Roman Empire, is NOT holy, nor Roman.

    Is Slashdot a slash and a dot?

  4. 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.

  5. Re:Riemann zeta function on Wikipedia by amorri09 · · Score: 2, Informative

    Wow, how wrong are you.....I honestly think that Numb3rs is the most contrived POS on TV. First off, The acting ISN'T good. Second, The plots are always work back kind of solutions packed with the mathmatical equivelant to the techno-babel you see on most network Sci-fi tv shows (eg. Hey! I was just reading about this thing called the Riemann Zeta something, lets make it into an episode that most likely has NOTHING to do with the proof or application of the proof itself...). Third, the plots of the show are amazingly un realistic.....like applying pattern algorithims that take into account 200 varibales to figure out what house a kidnapper is hiding in... Come On! Sorry, maybe im biased, but that show does nothing except get under my skin

  6. 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/

  7. 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.

    1. Re:You miss the point by Anonymous Coward · · Score: 1, Informative

      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. From the article:

      Booker commented that, "This work was made possible by a combination of theoretical advances and the power of modern computers." During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results.
  8. Re:question by l2718 · · Score: 3, Informative

    Now what the heck are degrees of those L-functions?

    This is where things get technical. The Riemann Zeta-function $\zeta(s) = \sum_n n^{-s}$ has the Euler product representation $\zeta(s) = \prod_p \left( 1 - p^{-s}\right)^{-1}$. Similarly, the Dirichlet L-functions $L(s;\chi) = \sum_n \chi(n)/(n^s)$ have the Euler product $\prod_p L_p(s;\chi)$ with $L_p(s;\chi) = 1/( 1 - \chi(p)/(p^s))$. In both cases, the factor at each prime $p$ takes the form $1 / ( 1 - a(p)/p^s )$, for some number $a(p)$ depending on $p$. We think of this factor as a the inverse of a polynomial of degree 1 in the variable $p^(-s)$ (the polynomial is $P(T) = 1 - aT$).

    Similarly, to GL(3) Hecke-Maass forms such as the ones computed by Booker and Ce Bian, there is an attached L-function $L(s;f)$ which can be represented as an Euler product, $\prod_p L_p(s;f)$. This time, however, the local factors $L_p(s;f)$ are the inverses of cubic polynomials, that is $1/L_p(s;f)$ takes the form $P(p^-s)$ where $P(T) = 1 - aT - bT^2 - cT^3$ for some coefficients $a,b,c$ depending on $p$ (and on $f$, of course). This is why we call it an L-function (or Euler product) of degree 3.

    Using the Fundamental Theorem of Algebra, it is common to factor the polynomial $P(T)$, and write it in the form $\prod_{j=1}^{3} ( 1 - \alpha_j(p) T)$. Thus an Euler product of degree $d$ takes the form:

    \prod_p \prod_{j=1}^{d} 1/(1-\alpha_j(p) p^{-s})
  9. Re:I have already solved this! by Kjella · · Score: 2, Informative

    I've long since forgotten the details, but there is a "proof" along the lines of his previous proofs that is simple, elegant and wrong. Most likely that was his proof and when he realized the flaws he never published it, so all that's left is an overly excited comment in a margin. Of course, that's an incredibly boring and everyday explaination, so it's usually discarded in favor of mystery and legend.

    --
    Live today, because you never know what tomorrow brings
  10. Re:question by MrSniffer · · Score: 2, Informative

    The "degree" is defined in this brief overview of the math, shown using conventional notation. http://www.aimath.org/news/gl3/technical.pdf An overview of this result can be found at this page http://www.aimath.org/news/gl3/