What Atiyah said (and didn’t say) about the Riemann Hypothesis
After a few days of suspense we finally had the chance of seeing Sir Michael Atiyah’s proof of Riemann Hypothesis (RH). As I said in a previous post, there was a lot of excitement the days previous to this event, but also some skepticism: though Atiyah is one of the best mathematicians of our times, the RH is a well-known problem that has eluded the minds of all the best mathematicians for 160 years. Following Atiyah’s talk, the excitement has faded and the skepticism has grown.
Since the RH is a very deep result, wrapped in complex technicalities, I was expecting the presentation to be quite dense, difficult to follow and in particular very fast-paced. After all, Atiyah was planning to show the proof of the most difficult problem in mathematics in just 45 minutes, the same time that I have exhausted many times to present much more trivial results. However, the presentation was the exact opposite of it: it was cheerful and light, with a long discussion about the history of the problem, abundant references to the work of other mathematicians, and even a movie recommendation.
Atiyah tells us in his talk that he stumbled across RH when he was developing a model for the fine-structure constant, a physical constant that Arnold Sommerfeld introduced in 1916 when he was extending the Bohr model of the atom. He also tells us that the key ingredient in his proposed model was the Todd function, a “magnificent animal” that he found in the writings of John von Neumann. “This Todd function is very well worth studying”, he tells us before jumping into some of its properties, and judging from the role it plays in RH this is something probably we all should do.
The problem is that if you google “Todd function” you won’t find too much information, so you have to rely on what Atiyah is telling you: that it is a weakly-analytic L2 function, analytic on compact sets, and polynomial on convex compact sets. That it is constructed from infinite iterations of exponentials and that it transforms Euler equation to Euler-Hamilton equation for quaternions. Not much more is said.
And then, at minute 36 we see a slide that contains the full proof of RH: Assuming that there is a zero not aligned in the critical line (but inside the critical strip) and applying the Todd function on Riemann´s zeta function, we make some simple manipulations, and voilà, we arrive to a contradiction. It takes him less than four minutes to go through the whole proof. Atiyah is confident that his result is correct and answered with a firm “yes” when asked if he thought his solution has the merit to win the Millennium Prize from the Clay Mathematics Institute.
Some of the skepticism around Atiyah’s proof has focused on two irrelevant points: that the proof is way too short for such a difficult problem and that some of his recent work in other areas has proved to be wrong. But as any mathematician knows neither the length of a proof nor the credentials of the author play a role in the validity of a theorem. The real skepticism arises due to the obscurity of the pillars that support his proof, which seem to appear only in other work that he hasn’t published yet. The properties that Atiyah gave of these Todd functions are somehow strange, but what is more puzzling is that you can’t find any mention of them – despite the claim that they appear in the works of von Neumann in the 1930s.
From the presentation alone it is not possible to conclude that Atiyah proved RH, and a thorough discussion – in the form of a publication, for example – is necessary. The night before the talk it started to circulate through emails what seemed to be a short draft of a paper written by Atiyah of his proof of RH, and it is now widely available online. It certainly contains all the material that he discussed over the talk hinting that he might really be the author.
The problem with this draft though is that you can’t say that it is “official” in the sense that you can’t find it in an official web page of Atiyah, or in a reputable repository of papers like arXiv. As such, this is a document that could be anything from an early draft of the paper, to a note not intended for distribution. And this is important, because no expert is going to review or comment a paper not knowing if it is the real thing.
It is possible that, just as I write this entry, a few referees of some high-profile journal are reviewing a paper with Atiyah’s proof of RH. If this is the case and they approve the paper for publication, the mathematical community will regard that as the necessary stamp of approval and we will all celebrate the achievement. On the other hand, if such paper is submitted but rejected and/or no official version of it is ever presented publicly, I’m afraid this is going to be the last time we heard anything from this episode.
Since the announcement of Atiyah’s talk in Heidelberg, we have seen all sorts of tweets, and memes, and Facebook posts, and blog entries (like mines) about him and about the RH. And there have also been trolls mocking the mental health and age of Atiyah or attacking the spirit of the Heidelberg Laureate Forum, and condescending voices of people that think that due to respect with the great mathematician we should just turn the page of what they believe was an embarrassing event. All this, of course, is just the price that is paid for doing things under the public scrutiny in these super-connected times in which we live.
I personally don’t think this is bad at all. Just for a change, it has been exciting to see thousands of people shifting their attention of the usual stream of news and following this story, talking about mathematics and trying to understand a little bit more of that beautiful problem that is the Riemann Hypothesis.
Following the two recommendations of my last entry, I would like to add to the list Harold Edwards’ “Riemann’s Zeta Function”, which can be found in a very nice and inexpensive Dover edition. This is a good book for anyone that has taken an Analysis course and wants to learn a bit more of this subject.