Terry Tao has a specific objection on his web page. http://terrytao.wordpress.com/2008/02/07/ structure-and-randomness-in-the-prime-numbers/#comment-30714

2 July, 2008 at 6:28 pm Terence Tao 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.

Li has fixed this problem, and is up to version 4 of his paper. The discussion on the Tao blog has continued, raising another problem. At least one correspondent says "this kind of proof can't work because of xxx". I guess we'll have to wait a few more days for the dust to settle. One thing consipicuously missing from the discussion is any kudos to Mr Li, or suggestion that he's developed any valuable ideas, or a new approach. Rich ---- Quoting Jason <jason@lunkwill.org>:

...

I had lunch with Sarnak today -- he looked at the paper and said that there are many problems with the proof.

Ah yes, proof by personal communication[1]:

proof by eminent authority: 'I saw Karp in the elevator and he said it was probably NP- complete.'

proof by personal communication: 'Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication].'

1: http://www.princeton.edu/~sacm/humor/proof.html

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Well, I guess everyone knows that after errors were pointed out by Tao, Connes, and ("I read in Wikipedia that some dude says he said so in the elevator, so it must be true") Sarnak, Li has withdrawn his eprint. I presume there is a causal connection. On Fri, 4 Jul 2008, Stephen Gray wrote:

Inspired by the idea of "proof by personal communication," I wanted to tell all of you that I just proved every outstanding conjecture in number theory, but unfortunately this note is too small to include them all.

This reminds me of a story I once heard about Hardy. (I'm telling this from memory so hopefully not mangling it--- the original source was probably a book by Paul Halmos.) It seems that Hardy was terrified of drowning and dreaded sea voyages. A colleague once received a postcard from Hardy saying he had proved the Riemann hypothesis and would write again once he had returned from a conference on the continent. Some time later, the colleague ran into Hardy, back in England, and inquired after the proof. Hardy blandly explained that, convinced that God's purpose in creating the universe was to put Hardy in it and then persecute him, he had concocted a diabolical plan to ensure his safety at sea. He explained that God was unable to sink Hardy's ship because He simply could not abide the idea that people might think Hardy really had proven the RH just before being sunk.