Xian-Jin Li and an attempted Proof of the Riemann Hypothesis
A new paper by Xian-Jin Li was uploaded on arxiv this week, with a claimed "proof" of the Riemann Hypothesis. Well, a "proof" is actually given for E. Bombieri's refinement of A. Weil's positivity condition, which implies the Riemann Hypothesis. The "proof" is said to be in the spirit of Alain Connes' approach to the Riemann Hypothesis.
Fortunately, (or unfortunately) it didn't take too long for Fields medalists to punch holes in the purported proof. Alain Connes himself, as well as Terence Tao pointed out certain problems on pages 20 and 29.
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).
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.
Xian-Jin Li has since posted revisions of his paper, specifically making changes on page 20 and 29, where Connes and Tao pointed out difficulties. However, it is doubtful Connes and Tao will take another look at the new version of the paper.