A couple of days ago, I wrote about Xian-Jin Li’s claimed proof of the Riemann hypothesis. Here are matters as they stand now:
Li’s preprint on the arXiv is now up to version 4. The initial problem identified by Terry Tao is now fixed; however a more serious problem pointed out by Alain Connes remains. Basically, Li’s paper appears to adelically integrate a function supported on the ideles, which cannot give anything useful because the ideles have measure zero in the adeles.
Terry Tao explains here why an approach such as Li’s, which basically only uses the multiplicative structure of the adeles, cannot work. Money quote:
The argument only uses the multiplicative structure of , but not the additive structure of k. (For instance, the fact that k is a cocompact discrete additive subgroup of A is not used.) Because of this, the arguments would still hold if we simply deleted a finite number of finite places v from the adeles (and from ). If the arguments worked, this would mean that the Weil-Bombieri positivity criterion (Theorem 3.2 in the paper) would continue to hold even after deleting an arbitrary number of places. But I am pretty sure one can cook up a function g which (assuming RH) fails this massively stronger positivity property (basically, one needs to take g to be a well chosen slowly varying function with broad support, so that the Mellin transforms at Riemann zeroes, as well as the pole at 1 and the place at infinity, are negligible but which gives a bad contribution to a single large prime (and many good contributions to other primes which we delete).)
Li would probably have been better off showing his work to some experts before releasing it on the arXiv. As any mathematician would testify, it can sometimes be surprisingly hard to realise your paper contains a mathematical error, especially if you are confident of its correctness.
Update(July 5): Li has now withdrawn his paper from the arXiv.