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A Step Towards Proving the Riemann Hypothesis
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."
Regardless of the money,
Ce Bian and Andrew Booker (and their computer) should at least win SOME prize and may need to practice their Sweedish.....
- Minutus cantorum, minutus balorum, minutus carborata descendum pantorum.
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.
enough room in the margin of this
text area to display it properly.
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.
(just smile and nod, smile and nod. they'll never know you have no idea what this means)
Absolute power corrupts absolutely. indymedia
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
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.
The popular T.V. Show NUMB3RS had an episode ("Prime Suspect") in which criminals kidnapped a child and demanded as ransom a possible proof of the Riemann Hypothesis from a mathematician. The proof would be used to steal interest rates from an encrypted website.
Fascinating!
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.
If they'd have left it alone in 1858 we wouldn't be having this trouble. If it ain't broke, don't fix it!
Riemann zeta function is the "mother of all L-functions".
...
zeta(s)=sum(n=1, inf)(1*n^-s)
Dirichle L-function is defined as
L(f, s)=sum(n=1, inf)(f(n)*n^-s)
so when f(n)=1, Dirichle L-function becomes Riemann zeta function.
L-function is just another representation (called Euler product) of Dirichle L-function.
L(f, s)=prod(prime p=1, inf) P(p, s)
where
P(p,s)= 1 + f(p)p^-s + f(p^2)p^-2s +
The Euler product I figured must work similar to the usual prime number decomposition: you got the sum of 1's and you got a product of primes.
That is how far I got.
Now what the heck are degrees of those L-functions?
I do not believe in karma. "Funny"=-6. Do good and forbid evil. Yours, Oft-Offtopic Flamebaiting Troll.
A proof of the Riemann Hypothesis itself won't have any effect on the security of encryption (if it did, you could compromise the encryption by just assuming the hypothesis is true and your exploit would work in nearly all cases). The only concern is if the process of developing the proof leads to an insight about the nature of prime numbers that weakens encryption in some other manner, but this wouldn't be the result of the Riemann Hypothesis itself.
...before 1859, when cars were pulled by horses and the Riemann hypothesis was still not unproven. Those were they days, I tell you, those they were.
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities
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/
You're thinking of physicists, who can prove this hypothesis for all prime numbers which are perfectly spherical and exist in a perfect vacuum.
how to invest, a novice's guide
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.
Umm... not to boost your ego too much, but if that seemed intuitive, you might consider reading ahead in your text books and looking at what else seems intuitive. If you manage to run your intuition through 3 var calculus, go for point set topology next. This "intuition" you speak of might be more of a gift than you realize. If you get through topology on your own, mention it to a professor who is known to be good at explaining things (those are usually the ones who actually understand and LIKE to explain things). He'll tell you what's next. You might have the ability to naturally translate between geometrical and symbolic view. Believe me, it's not common. And things you can learn to do with it are pretty cool.
Any guest worker system is indistinguishable from indentured servitude.
This article is related to Sage ( http://www.sagemath.org/ ), a free open-source math project. The article is about a computation (not using Sage) of an L-function, a computation about that L-function (using Sage), and a major new NSF-funded initiative to compute large tables of modular forms and L-functions that William Stein (director of the Sage project) is co-directing, which will have a large impact on Sage development.