During the last few days, there has been a flurry of activity following an announcement by Xian-Jin Li of a supposed proof of the Riemann Hypothesis, perhaps the most famous unsolved mathematics problem (now that Fermat’s Last Theorem has been solved.) Atle Selberg, an eminent mathematician, recently said:
If anything at all in our universe is correct, it has to be the Riemann Hypothesis, if for no other reasons, so for purely esthetical reasons.
On July 1, 2008, Li posted his proposed proof on arxiv.org here, stating in a short abstract:
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.
I recently noticed a quip that there were two sorts of mathematics papers – those where you can’t understand anything after the first page and those where you can’t understand anything after the first sentence. I’m glad that I’m not alone.
Subsequent events showed an interesting role for blogs even in a mathematical field where there are only a few people in the world who can read the paper.
Terence Tao observed the next day (July 2) on his blog (as a comment not even a thread):
It unfortunately seems that the decomposition claimed in equation (6.9) on page 20 of that paper is, in fact, impossible; it would endow the function h (which is holding the arithmetical information about the primes) with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry to make the adelic Fourier transform far more powerful than it really ought to be for this problem.
During the next couple of days, Li put two revised versions of his preprint onto archiv.org and seemed to feel that he could cope with this problem.
On July 3 at another blog, Fields Medalist Alain Connes stated:
I dont like to be too negative in my comments. Li’s paper is an attempt to prove a variant of the global trace formula of my paper in Selecta. The “proof” is that of Theorem 7.3 page 29 in Li’s paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).
On July 6, it was announced on arxiv.org that the paper had been withdrawn.
This paper has been withdrawn by the author, due to a mistake on page 29.
Li’s failed proof was 40 pages of dense mathematics, with one flaw on page 29. But the flaw was enough to cause the paper to fail.
Just think how much easier it would have been for Li had he published in Nature. His article would have been 3 pages long – a little section on the colorful history of the Riemannn Hypothesis, moving quickly to the “results” and implications. Maybe Nature editors would suggest that he mention the number of zeros was unprecedented. Due to “space limitations”, there would obviously be only a few sentences in the running text on how his proof actually worked, but the words “rigorous” and/or “conservative” would almost certainly have been used.
Their running text in Nature would perhaps said:
“We used the E. Bombieri’s highly conservative refinement of A. Weil’s rigorous positivity condition, which implies the Riemann hypothesis for the Riemann zeta function.”
Isn’t that much more convincing than Li’s original? Now you can see the realdefect in Li’s proofs: the failure to use either of the terms of “rigorous” or “conservative”. Instant climate science rejection. Had he correctly used the terms “rigorous” and “highly conservative”, nothing else would need to have been said and the paper would, of course, become an instant classic in climate science. Nobody would actually read it past the abstract, but it would be highly cited.
Had he published in Nature, if asked to provide details on his proof, Li could have refused and been backed up by Nature’s editors, who would have said that their policy does not require authors to show minute details of their methodology. If anyone doubted Li’s proof, if they were so smart, why didn’t they prove the Riemann Hypothesis themselves?
Perhaps 6 years later, someone would decode entrails of the Li “proof” from snippets, whereupon Li would haughtily announce that he had “moved on” to a new proof using an even more rigorous, conservative (and opaque) methodology and so he had been right all along.