Thursday, July 03, 2008

Xian-Jin Li: a wrong proof of the Riemann hypothesis

Update, July 6th: The paper has been withdrawn due to the error on page 29, the mistake pointed out by Alain Connes and described in the following July 3rd text:

Some readers who are interested in maths may have noticed a preprint by Xian-Jin Li,
A proof of the Riemann hypothesis.
It looks serious and it had to be a lot of work but it is not infinitely hard to see that it is wrong, especially if top mathematicians help us a bit.

Open the paper on page 29. Theorem 7.3 is formulated and a proof is offered there. It is a theorem similar to one of Alain Connes' theorems.

Alain Connes

The bug appears right below the equation (7.13) on page 29 where the author says:
We extend "h" to a function on "A" by defining h(lambda)=0 for lambda not in "J".
After this crucial comment, the function "h" is integrated over the adeles "A" a lot.
See an introduction to p-adic numbers and adeles
That's too bad because "J", the ideles (essentially invertible adeles), are a measure-zero subset of "A", the adeles - much like sets of rational numbers are measure-zero subsets of real intervals. So it is like defining a function of the real numbers that vanishes for all irrational numbers. Such a function is clearly unnatural in calculus and you might have some problems with integration.

So the integrals really vanish and they cannot be too useful. He could try to develop some "Dirac distributions" on adeles but such an approach would probably fail, too. All the following formulae therefore either say "0=0" which is an inconsequential truism or "0=something_else" which would prove that that the paper is wrong, too.

Terence Tao

While Alain Connes probably jumped right to the page 29 because this is where the stuff he is the world's #1 expert is concentrated, Terence Tao probably read it more systematically. Tao found a related bug already on page 20. Over there, the author claims that a function "h" of the adeles (the big set) exists that can be written as a certain sum over functions of the ideles (the small subset) in equation (6.9) of older versions.

But such a decomposition doesn't exist. Functions of the adeles (the big set) contain much more "information" than functions of the ideles (the small set) - at least if you allow the information to be "decoded" by simple sums only. Only the larger functions of the adeles know everything about the primes while the author tries to squeeze the information into the ideles.

The precise bug pointed out by Terence Tao was "fixed" or "avoided" in newer versions of the preprint, by using another "h", but Terence Tao agrees that Connes' bug explained above is way more serious - related to the very essence of Li's method - and it hasn't been fixed and can't be easily corrected.

When you think about the essence of the bugs, they can be summarized in the same way in both cases:

Xian-Jin Li needed to use the adelic (large set) Fourier transform but he could only say some relevant things about the idelic (small set) functions. And he decided to remove this tension by denying the difference between the adeles (and functions on them) - that are needed to reconstruct the information about the primes - and the ideles (and functions of them) - whose functioning he can master. ;-) But these are objects of very different "size".

So his strategy converted the difficulty of the proof of the Riemann hypothesis to the gap between adeles and ideles and he couldn't cross the gap. I tend to think that equivalently speaking, he was trying to use integrals (or sums) over unmeasurable sets (such as k* or something of the kind?).

At any rate, I guess that the guys who have been working with adeles for years probably know immediately that some functions "smell bad" and they can disprove such papers within minutes even if it takes weeks or months to write them and even though Xian-Jin Li is no Garrett Lisi or Lee Smolin but generally a competent mathematician.

Click the famous names of Fields medal winners to get to their blog text.

Bonus: An interesting comparison of sociology of number theory and climate science was written by Steve McIntyre.

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