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Prominent Mathematicians Rebuke Recent Riemann Hypothesis Proof
Bryan writes "Xian-Jin Li's purported proof of the Riemann Hypothesis (reported on recently) has been rebuked by Fields Medalist Terence Tao. Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts." Relatedly, jim.shilliday writes "The proof cites and appears to be based in part on the work of the leading French theorist Alain Connes. A few hours ago, Connes posted a comment on his blog stating that the purported proof is so badly flawed that he stopped reading it."
From the summary:
Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts
Why? I'm probably missing something obvious, I'm not even a mathematician to start with, but...
I mean, we (the world) do want to prove it right (or wrong) one day or another, don't we?
My husband is a mathematician, and he gets emails weekly from crackpots claiming to have disproved the proof of Fermat's Last Theorem or having proven the Riemann hypothesis or whatever. You can submit anything to the ArXiv, this shouldn't have even been news in the first place until it was confirmed.
Warning: Apple/Nintendo fangirl. Likes her electronics cute & cuddly. May be rabid.
The comments made by Tao and Connes are the sort of comments one would make if the paper was irrevocably flawed. For instance, Tao notes that "the decomposition claimed in equation (6.9) ... is, in fact, impossible; it would endow the function h ... with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry ..."
In more simple terms: Partway into the paper the author proved something that is definitely false; he then relied on this false theorem to complete the proof.
It's possible that Tao is wrong in his analysis or that the rest of the proof is actually independent of the false theorem that it appears to depend on. However, it's reasonably likely that this proof cannot be repaired.
Since the work based on the assumption that the hypothesis is true is in itself valuable, it will still be used.
It's just that a proof, if found, will elevate who-ever finds it to the status of mathematical superstar.
Consider this, we are still finding proof of various of Einstein's theories, but work based on his has been of real value for decades.
Here's another example that makes me sound all clever because I know it.
Newtons equations, and his entire body of work, completely failed to explain how it is that the moon can orbit the earth while the earth orbits the sun, and we *still* don't have the equation to explain that bugger.
There are specific n-body solutions, I've written one myself, but a solution for the general case? Nope, never been done.
Louis Pasteur spent most of his life on that particular problem, as have many other prominent scientists, all to no avail. We found a use for Newtons work regardless, and Einstein extended it successfully, even with that glaring hole.
A learning experience is one of those things that say, 'You know that thing you just did? Don't do that.' - D. Adams
This is not all that bad.
Probably many slashdotters are familiar with the discrete Fourier transform (used in JPEG encoding, incidentally). The DFT for sequences of length n fits together nicely with the DFT for longer sequences whose length is a multiple of n. If you try to put all these DFTs for sequences of different length together in a certain way and combine them with the continuous Fourier transform, you end up with something called the adelic Fourier transform. (That's a little bit different from how it is described in the usual books, but it is essentially equivalent.)
Next, if n has many factors then most integers will share a common factor with n; the proportion of integers that do not have a common factor will be small. Connes's statement that 'ideles form a set of measure zero' is what you get from this by taking the limit for large n.
Suppose you have a sequence a_1,...,a_n, where a_k is zero whenever k has a common factor with n. If n has many factors then a_k is usually zero and so the DFT of the sequence will be small. The limiting version of this fact is that if a function is supported on the ideles, then its adelic Fourier transform is zero. Thus, adelic Fourier theory is useless for studying such functions.
Connes is probably right that this is a showstopper.
A failed proof can still be worth reading, if it has interesting proof techniques or novel math structures in it. For example, ring theory, algebraic geometry, and moduli spaces were (as I understand it) due in part to failed proof attempts for Fermat's Last Theorem.