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Science

Prove The Riemann Hypothesis And Make.Money.Fast 15

Posted by michael from the makes-my-head-hurt dept.
waimate writes: "The Riemann hypothesis is all about the fundamental nature of those mysterious individuals, prime numbers. Proving it will have ramifications for quantum mechanics and improve our understanding of the nature of the universe, but more importantly will now win you one million dollars. With that kinda money, who cares who's the president." I'm afraid this article loses me about halfway through, but it looks interesting. Any mathematicians in the audience want to take a crack at explaining the second half of the article? :)
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Prove the Riemann Hypothesis and make.money.fast More

Prove the Riemann Hypothesis and make.money.fast

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  • Riemann hypothesis? Excellent! I shall make a point of writing half the proof down in my diary just before I die.
    It'll fit in nicely with my claims to have formulated a working GUT (which, obviously, I wouldn't be able to fit in the margin).
    Marios

  • *Another* prize -- anyone wokring on an index? (Score:3)

    by Kris_J ( 10111 ) on Wednesday November 08, 2000 @10:29PM (#636904) Homepage Journal
    Cheap access to space, X-prize, this -- there appear to be quite a number of high dollar value science prizes out there. Is anyone working on a list of these prizes anywhere on the 'Net. Even better would be a list of groups trying to win them. I think I'd like to be part of something like that...
  • complex numbers--numbers made from a real part (an ordinary number) and a so-called imaginary part (a multiple of i, the square root of 1).

    When I went to school, i was the square root of -1.

  • And it still is.
    a (example) complex number looks like this:
    234+45i
    which breaks down to 234 (the "real" part) and 45i (the imaginary part, which is imultipled by a real number)

    This is exactly what he said - why do you have a problem with this?
    --

  • According to his post i was claimed to be the sqrt of *1* not *-1* .. isn't that the problem he had with this? Haven't read the article so I'm not sure what it does say ..

  • According to his post i was claimed to be the sqrt of *1* not *-1* .. isn't that the problem he had with this? Haven't read the article so I'm not sure what it does say ..
    Odd - in my browser, both the original article and his quoting of it in his own post had the '-' there. Maybe you have webbrowser problems?
    --
  • The Reimann Hypothesis, as I understand, can be explained as follows: Let z(x) represent the zeta function. It is known that z(x)=0 for all x that are negative even numbers (-2, -4, -6...) Those are Real numbers (no imaginary part.)

    It is conjectured in RH that the other solutions, the 'non-trivial' solutions, are all of 'real part 1/2.' That could be represented that:
    z(x+yi)=0 iff (y=0 and x= an even negative) or (x=1/2 and y is undetermined).

    I'm new to Number Theory, but I've got a lead on proving Goldbach's Conjecture through a non-ambiguous lower bound of solutions. After that, the RH is what I'd attack next... It's been the top target since Hilbert said so a century ago, and Andrew Wiles of the Fermat proof still concurs.

  • I forgot to add:
    It has apparently been proven that if for some reason the RH should be disproved, it is because some prime quartets would perchance land in a zero balance (real part?). However that would occur, don't ask me yet, I've not researched it.

    The RH is much like Goldbach's Conjecture in how easy it is to "see" it as the answer, but proving it to infinity is extremely difficult. One math text descibed math as a way "to make the invisible visible." To me, RH and GB are too visible, but we need a proof that could make them evident to the blind. (Tribute to Euler)

  • Interesting Situation Indeed! (Score:4)

    by Mad Hughagi ( 193374 ) on Thursday November 09, 2000 @06:16AM (#636911) Homepage Journal
    Up until this point I figured that the zeta function was just another one of those 'rarely used' functions in the back of my math tables!

    In essence, the point of the second half of the article shows how current research in quantum mechanics can be applied to give a sound reasoning to the fact that the zeta function produces these 'zeros' in the complex plane.

    Connes has proposed that if one could construct a particular quantum system it may very well have energies (remember that in quantum mechanics energy levels are discrete - not continuous!) that correspond to these zeros produced by the zeta function. For this to be truly groundbreaking however the quantum system would have to display all of the energy levels corresponding to the zeros but it would have to not have extra energy levels. I don't know how to go much further into the development of a quantum state space based on prime numbers, however. Perhaps someone more well versed in QM could explain that.

    The true beauty of this proof is that it would give prime numbers an actual physical significance. Like it says in the article this could result in greatly expanding our ability to deal with many complex phenomena on the quantum scale and give us insights into many other phenomena. I for one am going to be following this one as close as possible!

  • Andrew Wiles spent seven years holed up in his attic solving Fermat's Last Theorem. If the Riemann proof requires similar effort, that $1m prize works out to $143K per year. That might not be so bad if it were a salary, but the risk is that you don't get the answer and spend years earning zero.

    Now if anyone wants to offer $1m just for trying to solve Riemann or Goldbach, count me in!!!

  • It's an HTML problem. The author used a "soft" hyphen (­) instead of a "hard" hyphen.
  • 1. The reason the Zeta function is useful in this case is because if all of it's zeroes lie in 1/2+nj (where n is a positive or negative real number and j is the square root of -1), the right side of the complex plane, then this means that the distribution of the primes does indeed follow 1/ln(x). The Zeta function is merely providing tolerance limits, but nevertheless it also implies that the primes are random.

    2. So you know the Zeta function is useful for this but now the question is how to show that all these zeroes lie in that line? It seems that Alain Connes has developed a quantum chaotic model that has energy gaps that are equivalent in proportion to the gaps found between zeroes of the Zeta function in the complex plane. But now he has to verify that there are no additional energy levels that do not lie on that line. This is the part that he may have a lot of difficulty solving. Also the n-adic geometry that Connes is using basically means that he is doing cyclic arithmetic. In a 5-adic system, 5+3=3, 2+4=1, and so on. It's just like a clock...once you pass the number at the top of the dial you start again at the lower numbers. Even so, I don't understand why he is using this mathematical system to describe separate dimensions for each prime number!?!

    3. So what if he verifies that there are no additional energy gaps that violate the 1/2+nj line? Then it sounds like he needs to verify this model experimentally with some real system.

    4. If all this comes to pass then I guess the primes are proved to follow the distribution more or less.

    An interesting side note. It should be obvious that this looks sort-of like a problem in control theory.

    In control theory, you try to analyze a system for stability based on its transfer function. If the transfer function has a denominator that can be set to zero then the values that create that condition are labeled "poles."

    It seems like in this case we could treat the Zeta function as the transfer function.

    If the poles of a system lie on the right side of the axis of the imaginary plane then the system is considered unstable and the time response is supposed to increase exponentially with time or increase exponentially as a sinusoid with time.

    I'm not saying that these are necessarily the same problem but it is interesting to consider the analogy. So is the tolerance of the logarithmic distribution of primes equivalent to the exponential time response of an unstable control system??? I guess that's what makes the distribution random - eh?

  • Actually, the zeta function is equal to the sum from n=1 to infinity of 1/(n^s) where s = x + iy is a complex number. If you let the imaginary part, y, equal zero than any calc student (hopefully) can tell that this infinite sum diverges for x leq 1. All can be saved though because there is a way to analytically continue the zeta function, except at x = 1 I believe. So someone proved that the analytical continuation of the zeta function had zeros where y=0 and x=-2k for k a postive int.

    So when someone tries to solve this function along the line x = 1/2, they are also working on this analytical continuation. I have looked at this and boy is it confusing.

    Man, this would be so much easier to type in tex. :)
  • I think it is... maybe?
    Right over my head, thats for sure.
  • *SIGH*
    Obviously I need to uncheck my +1 box more - yet another point of karma gone due to someone deciding my DEFAULT score is overrated :+(

    I wonder if I should start a new account to make my postings from, and hold on to my last few Karma above the 100 mark.....
    --

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