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