Last night a preprint by Xian-Jin Li appeared on the arXiv, claiming a proof of the Riemann Hypothesis. Preprints claiming such a proof have been pretty common, and always wrong. Most of them are obviously implausible, invoking a few pages of elementary mathematics and authored by people with no track record of doing serious mathematics research. This one is somewhat different, with the author a specialist in analytic number theory who does have a respectable publication record. Wikipedia has a listing for Li’s criterion, a positivity condition equivalent to the Riemann Hypothesis.
Li was a student of Louis de Branges, who also had made claims to have a proof, although as far as I know de Branges has not had a paper on the subject refereed and accepted by a journal. He describes his approach as using a trace formula and “in the spirit of A. Connes’s approach”. Li thanks
J.-P. Gabardo, L. de Branges, J. Vaaler, B. Conrey, and D. Cardon who have obtained academic positions in that order for him during his difficult times of finding a job.
but it is a little worrisome that he doesn’t explicitly thank any experts for consultations about this proof. If the arXiv submission of the preprint is the first time he has shown it to anyone, that dramatically increases the already high odds that there’s most likely a problem somewhere that he has missed.
I’m no expert in this subject, so in no position to check the proof or to have an intelligent opinion about whether his method of proof contains a new, promising idea. I suspect though that experts are already looking at this proof, and it appears to be written up in a way that should allow them to relatively quickly see whether it works. Given the history of this subject, I think the odds are against Li, but I’m curious to know what experts think of this.
This also has appeared on Slashdot. If your comment is like any of the ones there, please don’t submit it, but comments from the well-informed are strongly encouraged.
Update: It looks like a problem with the proof has been found. Terry Tao comments on his blog
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.
Update: Another Fields medalist heard from: Alain Connes comments as follows on his blog:
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).
Update: The paper has now been withdrawn by the author, “due to a mistake on pg. 29”.
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If you will accept a comment from a competent mathematician who is NOT an expert in analytic number theory, Li’s claimed proof of the Riemann Hypothesis looks like it might be the real deal. First, it is impressive that Bombieri and Lagarias thought enough of his positivity criterion to generalize it. Second, his paper is clearly written, and in fact serves as a good roadmap into various arcane developments in number theory such as adeles and ideles. Third, his paper follows a program, first enunciated in the 1970’s, for approaching the Riemann Hypothesis by studying the reals and the p-adic numbers simultaneously in one large entity, the direct product of all of these. Fourth, he clearly understands the deep relationship between the Mellin transform, the Riemann zeta function, and the invariant measure dt/t on the positive reals. In fact, the measure dt/t is Haar measure on the group of positive reals under multiplication, the Mellin transform is the Fourier transform with respect to this invariant measure, and the zeta function is essentially the Mellin transform of the fractional part function, t – [t]. Fifth, there are some new ingredients in his proof, especially his positivity criterion and various things about special functions from Whitaker and Watson.
The only errors I could find are a misspelling of Dedekind’s name on page 2, and a few lapses in English of the sort to be expected of a non-native speaker. I’m rooting for him. I hope someone more expert than I will peruse his work and give a more detailed and expert opinion.
I’m sure that he’s no crank, but as the field stands, using ideles and adeles and the Mellin transform is pretty standard stuff in the area. The proof looks awfully elementary to me. (I’m not an analytic number theorist either.) It would be great that if a famous open problem like the Riemann hypothesis had a proof sufficiently elementary that I could understand it, but I would be pretty surprised.
JA, love your blog!
And most urgent: ditto Chip. I had the same initial thought as several others when I realized that Li is a former Ph.D. student of Louis de Branges, but after looking over Li’s papers– which have been published and cited— it seems to me that they appear to fit into a community effort involving some of the best mathematicians of our time, most notably Bombieri and Lagarias (who seem to have generalized a RH criterion due to Li). While I don’t know much about this area, I recognize (or think I do) some ingredients which leading mathematicians have said might play a role in a proof of RH. A genuine proof of a long open conjecture which uses familiar ingredients in ways which had somehow been previously overlooked would be VERY surprising (de Branges’s proof of Bieberbach conjecture is said to fit that bill), but not unprecedented.
So I think this might actually be the real deal! Wow, if so!
Er.. sorry, Peter, thought I was posting in John Armstrong’s blog. (Embarrassed grin.) But you have a great blog too!
My impression is that de Branges’ proof of the Bieberbach Conjecture was very technical, but that later someone found a more elementary (and shorter) proof.
Terry Tao thinks it’s incorrect:
Thanks Ian, I’ve added that to the posting.
Xian-Jin Li has posted a new version (actually two new versions!) of his preprint on the Arxiv. Most pertinently, the definition of the function h on page 20 has changed; so perhaps this addresses Tao’s objection on his blog.
The reason I will be very surprised if this proof turns out to be correct is that it involves mostly functional analysis on the adeles. It has been generally believed that such techniques are not sufficient to prove Riemann. It would be a stunning achievement indeed if Riemann is solved using only such elementary tools!
Apparently the half of the paper (at least) is “copied”
from Alain Conne’s paper cited in reference, especially the
part metionned by Terence Tao. So I guess Connes would
be best placed to evaluate this paper. For instance the definition
on which Tao focuses on corresponds to eq. 18 p42 in Connes’
Also, it’s really a detail, but I was really surprised that in this paper,
which seems to be quite a serious work (especially compared
to previous papers published in the last months…), the reference
to Weil famous paper (ref 15) serves only to reference the
definition of the Schwartz-Bruhat space….. Even Connes in his
paper references Bruhat paper, which is much more relevant!
Weil paper has probably much more to do with RH, but certainly
not with the definition of the Schwartz-Bruhat space….
Connes has commented on his blog
Thanks, I added the information about Connes’s comment to the posting.
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And now we have version 4!
Indeed. I checked the places where Connes and Tao pointed out, and that’s where he made the chages.
P.20, the definition of the formula 6.9 (Tao)
p.29, the first paragraph of the theorem 7.3 (Connes)
I didn’t look anywhere else.
After reading the comments by Tao and Connes, my enthusiasm has considerably diminished :-/
Actually I had first a similar doubt as Tao had, but
comparing a bit more closely Li and Connes papers,
I am now more confident in this part of the paper: actually
this part is almost copied and pasted from Connes paper,
and if it were wrong, it may also possibly be wrong
in Connes paper (assuming I have not missed a difference
between the two papers).
However the point raised by Connes is really a major objection
to Li’ results (as for me)…
Terence Tao made a further comment on problems with the proof. He also explaines in more detail what Connes pointed out.
That was Tao’s earlier comment about the paper. Perhaps you’re thinking of this comment:
It will be interesting to know the opinion of Xian-Ji Li himself.
Has Li something to say to his “critics” ?
It will be very important to know also the point of view of
Prof. Xian-Ji Li.
Can anybody tell us anything on this?
Li hasn’t withdrawn his paper. He is very confident in it?
Regarding Connes, Kowalski, Silberman and Tao comments on the quotient space A/k^*, I recall a comment by André Weil that could be translated as “The search for an interpretation of [the idele class group] seems to me to be one of the most fundamental problems in number theory today; it is possible that one such interpretation holds the key to the Riemann hypothesis” (Sur la théorie du corps de classes).
Just found that Li had withdrawn his paper.
I’m glad that such pure mathematical research is discussed here. Seeing how proofs are checked and commented upon, and withdrawn if an error is found, makes it very interesting. It’s clear that there is a lot of discipline involved in step-by-step proofs of theorems in pure mathematics because it is possible to make an error which breaks the chain of reasoning.
Applied mathematics is far less risky, because where a model is being constructed a logical error can usually be corrected without the whole model collapsing. I think this the distinction between the kind of model building mathematics being done in string theory, and the formal proofs of pure mathematics.
Interesting to read about this case. Fortunately, I see no backfire as in the case of, as I remind, Penny Smith on the Navier-Stokes Equation problem.
It is also very interesting that the error was pointed out in the blogosphere…