One reader was curious if we had anything to say about the recent preprint by Xian-Jin Li entitled “A proof of the Riemann hypothesis”. Unfortunately, analytic number theory seems to be a weak spot of the mathematical blogosphere, so none of us seemed inclined to go through the paper and look for mistakes. Luckily, Terry Tao did and thinks he has found a mistake (which the author may claim to have fixed…things are starting to get a little confusing). Alain Connes also seems to be unconvinced. Oops.
Which leaves the rest of us to wonder what happened. I mean, this paper looked promising precisely because it didn’t look like the work of a crackpot. Li has a Ph.D. from Purdue (in mathematics) and is a mathematics professor at Brigham Young, and analytic number theory is his research area. He has several other unsuspicious articles on the arXiv, and the style of his Riemann hypothesis article is wholly unremarkable (considering that it claims to prove probably the most celebrated open problem still at large in the mathematical world). Why would someone risk the level of embarrassment involved in putting a proof of RH which had not been really thoroughly vetted on the arXiv, apparently with no warning (whether it can be fixed or not, if Terry Tao found a problem in less than 24 hours after it was placed on the arXiv, it definitely was not vetted thoroughly enough before being released on the world. It’s also on its 4th version on the arXiv in 3 days)? What was the hurry?
I can’t really speak to Li’s situation, since I don’t know the guy. It may well be that he sent his preprint to Tao and Connes and they didn’t get around to reading it. But if he didn’t, that was a huge mistake on his part, one which definitely makes him look more crackpotty than I expect he wants. If he didn’t give any conference talks on the subject before releasing the preprint, that was a huge mistake. Honestly, I think putting it on the arXiv, where it will remain forever, taunting him, rather than his personal webpage was something of a mistake. After all, you want a chance to get comments from the people who might be able to point out any mistakes you made before you end up on Slashdot. While this goes double, or perhaps n-uple for some large n if trying to prove an important problem like RH, I think it’s a good point in general that you should tell people about your work while it is still in its formative stages. It could save you a lot of pain. Admittedly, some people worry about being scooped, but I feel like this is the sort of thing that people are naturally more paranoid about than they should be. Ultimately, it would be better if we shared all our good ideas. After all, if somebody else does something cool with an idea you had, that just makes you look smarter for having such a good idea.
[Ed. – I changed the title of this post, since the original one was a bit more inflamatory than I intended]