Sticking with the theme of the Riemann Hypothesis, the AMS has recently posted some articles to appear in an upcoming issue of the AMS Bulletin, one of which contains a long interview with Atle Selberg, who died last summer at the age of 90. Selberg had been a professor at the IAS and an expert in analytic number theory, responsible for some of the most important developments in the subject during the 20th century. A large part of the interview concerns in one way or another the Riemann Hypothesis, which is a central concern of Selberg’s mathematical research, with his work on it beginning during the German occupation of Norway when he was still a student, Some thoughts from Selberg on the subject:
“If anything at all in our universe is correct, it has to be the Riemann Hypothesis, if for no other reasons, so for purely esthetical reasons.” He always emphasized the importance of simplicity in mathematics and that “the simple ideas are the ones that will survive.”
About whether there is a spectral problem that gives the zeros of the zeta-function, useful for proving the RH:
That is certainly a thought that several people have had. In fact, there have been some people that have been able to construct such a space, if they assume that the Riemann hypothesis is correct, and where they can define an operator that is relevant. Well and good, but it gives us basically nothing, of course. It does not help much if one has to postulate the results beforehand—there is not much worth in that.
About his own attempts to find a proof:
Once I had an idea that I thought perhaps could lead to a proof….
[gives some details]
After a while I became more and more convinced that it would not work as I had thought initially. It just seemed unlikely to me. However, I have now and then seen that people have attacked a problem in a way that seemed “hare-brained”, to use an English term, but then it turned out that they could make it work. They have proven something that would not be easy to prove in another way. On the other hand, I have seen people have ideas that seemed absolutely brilliant, but the only problem is that if one follows these to the end one is not able to get anything out of it after all. So it works both ways: sometimes a good idea does not work, and what seems like a bad, even idiotic idea, may actually work.
About Connes’s work on the RH:
Yes, that is a new way to arrive at the explicit formulas—a new access, so to say—but it basically does not give more than what one already had. Connes undoubtedly believed to begin with that what he was doing should lead towards a proof, but it turned out that it does not lead further than other attempts. When I last talked with him he had realized this. This often happens with types of work that are rather formal. There was, for example, a Japanese mathematician, Matsumoto, who gave several lectures that made quite a few people believe that he had the proof.
I think it is a good possibility that it will take a long time before it is decided. From time to time people have been optimistic. Hilbert, when he presented his problems in 1900, thought that the Riemann hypothesis was one of the problems that one would see the solution of before too long a time had elapsed. Today it is a little more than one hundred years since he gave his famous lecture on these problems. So one must say that his opinion was wrong. Many of the problems that he considered to be more difficult turned out to be considerably simpler to solve.