posted
A paper(http://arxiv.org/abs/math.NT/0703367) was just released claiming to have disproved the Riemann Hypthesis (RH).

This paper is unique in that among papers claiming to prove/disprove it in that

a) it actually has math in it

and

b) it's not immediately obviously flawed.

I just read through it myself, but with my weak background in analysis and number theory, I can't be sure of all the claims made. Hopefully in a bit, we'll get some real mathematicians figuring out if it's officially disproven or not.

For those who care, the Riemann Hypothesis is one of the millennium problems (a la Poincare Conjecture recently in the news. Unfortunately, disproving RH doesn't net one 1 million $ as proving it would). Itcan be stated as follows:

Consider f(z) = 1^(-z) + 2^(-z) + 3^(-z)+ ...

where z is any complex number.

Pick a particular complex number c and assume f(c) = 0. Then either the real part of c is a negative integer, or the real part of c is 1/2.

RH is important as it tells us something about the distribution of prime numbers - which is very important in some popular encryption schemes (among other important uses).

posted
Hmm. I seem to recall that Riemann has some extremely important implications for the more obscure branches of quantum mechanics, but I don't offhand remember what they are. It does seem to me that any complicated existence proof is rather taking the long way around, though - just give us the counterexample!
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posted
You must be a physicist. Who need to know what the counterexample is when you can prove that one exists?
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quote:Hmm. I seem to recall that Riemann has some extremely important implications for the more obscure branches of quantum mechanics,

The Montgomery-Odlyzko Law: The distribution of the spacings between successive non-trivial zeros of the Riemann zeta function (suitably normalized) is statistically identical with the distribution of eigenvalue spacings in a GUE operator.

Apparently, this means there is some connection between the distribution of prime numbers and the distribution of eigenvalues for Gaussian-random Hermitian matrices, which are used to model certain quantum-dynamical systems.

This is from "Prime Obsession" by John Derbyshire. I have no true idea what it means, though.
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"By using Fourier analysis on number fields, we prove in this paper E. Bombieri’s refinement of A. Weil’s positivity condition, which implies the Riemann hypothesis for the Riemann zeta function in the spirit of A. Connes’ approach to the Riemann hypothesis."
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posted
Extremely important, if correct. Arxiv doesn't list a journal submission, but these things take time, that's why arXiv exists in the first place.
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posted
Yeah, there's a lot of skepticism about it because it's not yet reviewed. I don't even understand the single-sentence summary.
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posted
There is also skepticism because the gentleman's advisor has backed the wrong horse in this race on previous occasions.
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posted
So if someone has backed several previous Riemann proofs that turned out not to be, that isn't grounds for skepticism?
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posted
This isn't even news until it's been reviewed, so yes the assumption is that it's wrong--no matter who submitted it.

I have to say that I think this trolling of arxiv for possibly exciting results is a terrible misuse of the service.
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