Fourteen
Atle Selberg
Atle Selberg was one of the most distinguished number theorists of the twentieth century, having contributed over a long career to our understanding of the primes, through his work in analytic number theory, specifically his results on the Riemann zeta function and on sieves. In 1949 he produced an elementary, but by no means simple, proof of the Prime Number Theorem, which had previously required deep theorems from complex analysis. The mathematical literature abounds in references to his work—the Selberg sieve, the Selberg trace formula, the Selberg zeta function, the Selberg identity, the Selberg asymptotic formula, and so on.
At the International Congress of Mathematicians held in Cambridge, Massachusetts in 1950, he received one of the two Fields Medals given that year. Only two had been awarded previously, at the Congress in Oslo in 1936. So the importance of his work was recognized early on. He received his PhD at the University of Oslo in 1943 and was at the Institute for Advanced Study at Princeton continuously from 1949 until his death. In 1986 he won the prestigious Wolf Prize.
The following conversation was held in the offices of the American Institute of Mathematics in Palo Alto, California, in June 1999. A draft of the transcription was found in Professor Selberg’s papers after his death. We are grateful to Professors Dennis Hejhal and Peter Sarnak, both long-time colleagues of Professor Selberg, for their help in clarifying some elements of the text and to Professor Selberg’s family—his wife, Betty Compton Selberg, son Lars, and daughter Ingrid—for their help in filling a few gaps in the details of Selberg’s early years in Norway, as well as clarifying some stylistic infelicities in the earlier draft.
“I Essentially Had to Find My Way Entirely Alone”
MP: You have been at the Institute for Advanced Study for a long time, over fifty years, I believe.
Selberg: Well, I’ll tell you. I came in August of 1947 as a temporary member and I stayed for a year. Then I took a position at Syracuse. I thought it might be interesting to have the experience of an American university. While I was there I got an offer to come back to the Institute, so I returned there in August 1949. Come late August of this year, I will have been there continuously for fifty years.
Figure 14.1 A photo of Selberg from a Norwegian newspaper in 1949, the year in which he was made a permanent member of the IAS.
MP: So what are you now, a Norwegian mathematician, an American mathematician, a Norwegian-American mathematician?
Selberg: I think that in many ways I would characterize myself as a Norwegian mathematician more than an American one because I have a very different background. In a sense I found my way in an environment where I essentially had to find my way entirely alone, because even when I came to Oslo, there was no professor there who had interests similar to those I already had. The closest one in Norway would have been Viggo Brun, but he was at the Norwegian Institute of Technology in Trondheim. But even had he been in Oslo, I always had difficulties understanding the way he thought. I had tried even before I came to Oslo to read some articles that he had written about the sieve method. They had some interesting geometrical diagrams and figures and so forth, but I looked at them repeatedly, and they didn’t mean anything to me. I only understood about the sieve method much later when I got into it from a different side, when I first invented my own sieve method and reflected on the principle behind it. And then finally I understood his articles.
MP: Just Wednesday of this week, using Viggo Brun’s sieve, I finished for my undergraduate class in number theory the proof of the convergence of the sum of the reciprocals of the twin primes. And it took me three days to get through the argument. For undergraduates it’s a bit of a challenge.
Selberg: But you probably followed the way that Hans Rademacher did it.
MP: Exactly.
Selberg: You see, until Rademacher did this, I think hardly anyone understood it.
MP: I have never gone back to look at the original—and maybe it’s just as well.
Selberg: Brun had a very original mind. And he was a very nice man. I was very fond of him as a human being.
MP: How long did he live? I can’t remember.
Selberg: He died in 1978. I did a memorial speech for him at the Academy of Science and Letters in Oslo, in ’79. He was 93 years old. He was born in 1885. It was a good year for mathematicians. Hermann Weyl was born that year, and J. E. Littlewood. And there were probably some others.
MP: A very good year. I was a student of George Pólya here at Stanford, and I was thinking about something just the other day. I saw the film about S. Ramanujan—I was showing it to my number theory class, actually—and it occurred to me that Pólya and Ramanujan were both born in 1887 (thus making it not a bad year either!). It occurred to me that Pólya went to Cambridge and Oxford in 1924 to work with G. H. Hardy and Littlewood as a “young man.” But by the time he got to Cambridge, Ramanujan had been dead for four years. I knew Pólya for the last thirty years of his life so, had Ramanujan lived, I could have encountered Ramanujan at some point. But because he died so young, he seems to be someone from way back in history somewhere.
Selberg: That is true. Well, of course, Pólya was a very important figure in many ways. And he might have met Ramanujan elsewhere. Ramanujan was very important for me. I came across him by accident, in a way. There was a professor of mathematics in Oslo, Carl Størmer, who had started out in pure mathematics and had some results in number theory going way back. The results still carry his name, in relation to solutions of the Pell equation. Later he mostly investigated the Northern Lights. He still kept up an interest in number theory, and, probably spurred on by the appearance of Hardy’s book about Ramanujan, he wrote an article about Ramanujan in the periodical, the Nordisk Matematisk Tidskrift (“Tidskrift” is a literal translation of Zeitschrift—you don’t say that in English). This is a magazine directed mostly to teachers in the gymnasium and to students who have an interest in mathematics, also to some amateurs. In this article he quoted a number of Ramanujan’s results and wrote about his life. I saw it and found it extremely fascinating. I had been reading mathematics on my own; at that time I was probably sixteen.
Somewhat later one of my brothers [Sigmund], who was still at the University and interested in mathematics, already interested in number theory (diophantine equations), brought home at vacation time Ramanujan’s Collected Works, which he had taken out of the University Library. So I had an occasion to see that, which was very important. I did have some interest in number theory before, and that was also occasioned by that same brother who had pointed out to me in a book in my father’s library—my father had a quite large mathematical library—an algebra book by a Frenchman, J. A. Serret, which had a chapter about P. L. Chebyshev’s work on primes. It was written before the work by Jacques Hadamard and Charles de la Vallée-Poussin, but it had Chebyshev’s inequalities and such. So I read that, and it fascinated me very much. I must say I wasn’t interested in much of anything in the rest of that book.
MP: Yes, I’ve seen the book, and it is certainly not uniformly interesting.
Selberg: So I think these two things pretty much gave me direction. There was a third thing. In the middle 1930s there came the work of Erich Hecke in connection with the Hecke operators, bringing together the functional properties that lie behind the Dirichlet series and tying this up with the multiplicative properties. That I found very fascinating. I thought at the time that this would give access to things like the Riemann hypothesis. Things turned out not to be quite that simple. And they still haven’t become simple.
Figure 14.2 A happy lecturer—Selberg at Columbia University
Growing Up in Norway
MP: No, not at all. You mentioned that your father had a really good mathematical library. Was he a mathematician?
Selberg: Actually, not originally, but he became one. He was the youngest boy on a farm, and he stayed home after the older boys had left. He left when he was about twenty—before that he had only elementary schooling that he got in the countryside—and he tried to educate himself by taking some quick courses that led him to the final exam of the gymnasium in one year. Then he took some teacher training and worked as a teacher for a couple of years. After that he moved into banking for a while and later into life insurance, where he did very well. In 1920–1921 he was the head of the western Norway division of the largest life insurance company. But he had become very interested in mathematics. He didn’t like the work he was in and wanted to go back into education, so he went to the university to take exams from time to time. Mathematics was his favorite field. He finished the necessary education to become a teacher in the gymnasium with mathematics his main specialty. Later he got a doctor’s degree. He was then about fifty years old, with nine children.
MP: Nine children!
Selberg: I am the youngest. I had four brothers and four sisters. Now I have only one sister left.
Because he was not in a place where there was a good mathematical library, he had to buy the books for himself. He bought a lot of good things. His main interest was algebra, but he had lots of other stuff as well, function theory, collected works (Riemann’s, for example, and Abel’s, of course, and Gauss’s), and he also subscribed to Sophus Lie’s collected works that were coming out at that time, also the Enzyklopädie der Mathematischen Wissenschaften. Of course, as the years went by that took up a very large space! He ended up with a quite large library. He never tried to influence me to go into mathematics, so I picked it up on my own.
MP: Didn’t you have a second brother who was a mathematician?
Selberg: Well, I had my older brother Henrik, eleven years older than I, who actually worked in value distribution theory, in particular, for the so-called algebraic functions, those that are finitely many valued, also in statistics. He worked some in industry for the Nobel Company, which was mainly interested in explosions and detonations. He was very helpful to me when I came to the university; he was then already on the university staff, not as a full professor but something corresponding to an associate professor. And actually when I wrote my first paper in the late summer of 1935, the year I was coming to the university, he typed it out for me. And beyond that, because of my bad handwriting, I think, he then wrote in the formulas. At that time you couldn’t really do many formulas by typewriter; you wrote them in by hand. So he was very helpful in those ways, but we did not have the overlapping interests that my other brother and I had.
MP: When the family was growing up, where were you? You were not in Oslo, right?
Selberg: No, actually, the first place I can remember is called Voss, which is on the line between Bergen and Oslo, on the western side of the mountains. It’s the most substantial station before you get to Bergen. I was not born there; I was born in the South of Norway. I have no recollections of the place, and I’ve never even gone back to look at it, though it had some meaning for my older siblings. But I was living outside Bergen when I got my early schooling, elementary school. For middle school I went to Bergen by train. It was rather nice in a way, sort of a commuter train that took a little over half an hour. We did a lot of school work on the train. There were quite a few students commuting like that. Then the family moved to the eastern part. My father was really working in the insurance business. In 1930–1931 he was appointed principal (or rector as it was called then) in the school system in Gjøvik, in the gymnasium, which he held for a number of years. At the same time he had another position as inspector of the lower schools. This is something that was not usually combined, but I think it gave him a slightly higher salary, which he may have needed with all his children. He had taken a very serious loss in income when he left the insurance company. There had been a boom right around the time of the First World War, and then there came a downturn, particularly in shipping, and that affected Norway. I think he may not have realized how severe the loss in income might be.
Interested in Numbers at an Early Age
At any rate I became interested in mathematics without anyone in the family pushing me. I was, at an early age, somewhat interested in numbers. I remember I was out with some other boys playing a ball game, not soccer, but another ball game that was played particularly by children, somewhat related to the American baseball. It’s also somewhat different and is played with a much softer ball. And the bat is also somewhat different. But it had many of the features of baseball—bases and running and such. And during these games there are times when you stand around and don’t have much to do. I remember that by doing some mental calculations in my head I noticed that differences between the consecutive squares give you the consecutive odd numbers. I did it just by numerical examples; I didn’t know how to use letters. I found sort of a mental proof by interjecting between n2 and (n + 1)2 the number n(n + 1). Then you can easily find the difference on both sides. It made quite an impression on me; it was the first time I made a mathematical discovery.
MP: And this was at age . . .
Selberg: I think about ten. Later I started looking at some of the books in my father’s library. The thing that really started me reading mathematics seriously was a book that included Leibniz’s formula for π/4 = 1 – 1/3 + 1/5, and so forth. That seemed a very striking statement. So to find out how all of this hangs together, I decided I wanted to read that book. It was a set of lectures by Professor Størmer, whom I mentioned earlier. It was a large course in calculus with a little bit of complex variables. This was a rather strange thing to start with. The book started by defining the real numbers with the Dedekind cuts. It’s a wonder I didn’t give up right there, because I found it very boring. I couldn’t for the life of me see what was the use of it. I didn’t see any difficulty with the concept of a real number. I thought at that age that you can write anything down as an infinite expansion, and that gives you a number.
MP: So much for too much rigor too early!
Euler
Selberg: Yes. Take Euler. I think he thought that he knew perfectly well what a real number was. He could calculate with them; he could do anything. I don’t think he was in any danger of making mathematical errors because he didn’t understand properly what a real number was.
MP: I think I’m beginning to suspect that you would not have seen a future for yourself in foundations of mathematics.
Selberg: No! Certainly not. I was very happy when I finished with that chapter and then it started to become interesting. It was a very lopsided mathematical education, you see, because it meant that I met the trigonometric functions first in terms of the Euler formula, in terms of the exponentials. I had trigonometry in school much, much later.
MP: It’s interesting that you cite Euler. Some of my favorite mathematicians, when asked who their all-time hero in mathematics is, will cite Euler. And, of course, I would say that myself. His work just has enormous appeal, partly, as you say, because he doesn’t get tangled up in all sorts of details in foundations.
Selberg: He operated at times rather recklessly, in matters of divergence, and things like that. But he had good instincts. So he didn’t really go astray.
MP: He also had extraordinary taste in finding good questions.
Selberg: He was a remarkable talent and extremely prolific.
MP: There’s another aspect of Euler that I like. When one reads his work, he talks about the problem, why he got interested in it, how he may have gone down the wrong path for a while, backed up, and started something else.
Selberg: That was a different time. These days if everybody started to do that, I think the editors of the journals would. . . .
MP: We can’t afford it! But it does make more pleasurable reading.
Selberg: Oh, yes! That is true. It was much more human.
MP: Of course, it’s also nice to see that even a giant in the field could start off with the wrong approach and be willing to tell you about it. Gauss was not.
Selberg: Gauss, of course, had a wrong philosophy. One has the impression, or at least I have the impression to some extent, that Gauss, when he wrote his Disquisitiones, was thinking of leaving something for number theory like Euclid’s Elements, something that would stand for centuries. He does make a remark somewhere that he wouldn’t expose the things that, as in a building, went into the construction. He was hiding his tracks, as other people said. This is a nonsense attitude, because something could only remain in the position of Euclid’s Elements if mathematics stagnated for a long time. Of course, Euclid’s Elements remained the same; nothing new was being done anywhere. This was thought to be the last word; nothing more could be done. It’s obvious that as long as mathematics is thriving, the form that any specific paper or book or other form that mathematics takes is always advancing. The content remains.
Figure 14.3 The young Selberg family in the sixties—Ingrid, Atle, Lars, and Hedvig.
Ramanujan
MP: Let’s go back to your school years. As a boy you had gone through a number of the formulas of Ramanujan before you even got to Oslo, to the university.
Selberg: Well, yes, I started working on some of his formulas. I wrote a paper that I submitted that same fall that I entered the university. It was in German, the foreign language I knew best at that time. It was on some arithmetical identities, some connections between series and infinite products and continued fractions. It was published the next year. I had sent it to G. N. Watson—I think he was in Birmingham—who was the expert at the time on Ramanujan’s Nachlass. It took him a long time to send it back, and I was rather impatient, I must say. Later, when I had to referee papers, I came to understand how that happens.
MP: But when you got to Oslo, your dissertation ended up being on the Riemann zeta function.
Selberg: Well, you see, I had gotten interested in mathematics through the work of Ramanujan and then later through some of Hecke’s work in modular forms. So I was reading some number theory, specifically the Cambridge Tracts, like the one by A. E. Ingham. I never liked very fat books, so these Cambridge Tracts were the kind of books I could get something out of. I had started some attempts at work on the zeros on the critical line, using some ideas that were a bit different from the ones that had been used before. I had some results, but they didn’t go so far as Hardy and Littlewood’s. I was looking at their paper, particularly some end remarks they made—why they couldn’t get more out of their method, and they gave some sort of explanation for that. I thought about that a bit, and then I realized that was really pure hogwash. That was not the reason that it wouldn’t work. Basically it has to do with variations of arguments in small intervals or so. But it was not the argument, it was the amplitude, the absolute value, that fluctuated too much. So then I thought of introducing a damping factor that would dampen the fluctuations. I experimented first with very simple forms of the auxiliary factor that should dampen the fluctuations, but one that did not introduce any sign changes or anything. I gradually improved it, to see if I could get a better result, then I made a guess, and it gave me the final result. That gave me the dissertation.
No Dissertation Advisor
MP: Who was your dissertation advisor, even nominally?
Selberg: I didn’t have any. We didn’t have that system in Norway. You see, you wrote a dissertation entirely on your own. If you thought you had something good enough, you turned it in to the university, and a committee took a look at it. You didn’t have to have any connection with the university to do that. You didn’t even need to have a university degree in advance. Anyone could do it.
MP: We certainly don’t give degrees unless people pay tuition and lots of it!
Selbeg: I know.
Figure 14.4 Selberg celebrating being named a Knight Commander of the Royal Order of St. Olav in 1987 by the King of Norway with his daughter Ingrid, wife Hedvig, and son Lars.
MP: It’s a different system entirely.
Selberg: Besides the dissertation and the disputation in relation to that, you also have to give some lectures, one on a subject that you choose yourself and one on a subject given to you by the committee. I don’t know whether that has been modified, but when I got my doctor’s degree, that gave me the right to lecture at the university and the university had to provide me with the auditorium. On the other hand they didn’t have to pay me! In Germany, there were lots of people with doctor’s degrees who were not paid by the university but were paid by the students.
MP: The privatdozents.
Selberg: Yes. We didn’t have them in Norway, but in theory, there could have been. I don’t think anyone could have made a living at it.
MP: There were a lot around Göttingen who were just barely making a living at it. It was a hard life.
Selberg: Oh yes.
Prison Camp
MP: Now, what did you do immediately after your doctorate?
Selberg: I got my doctorate during the War, and I was a research fellow at the University at that time. But actually the University was closed in the fall of the next year, in the fall of 1943, not long after I had my disputation and got my degree. I was actually for a while in prison because I happened to be in the University at the time the building was occupied by the Germans. But I was let out relatively early because I had friends in high places. There was a professor of actuarial mathematics with whom I had reasonably good relations before the War came, before the occupation. I had on some occasions actually helped him on some mathematical questions. He got involved in the Quisling party, and I think that it was his influence that got me out of this camp.
MP: How long were you in the prison camp?
Selberg: Not long, not more than one and a half months. Actually, about the same amount of time that I had been a prisoner of war in 1940. When I was released from the camp one of the conditions was that I shouldn’t be in Oslo. So I went to live where my parents were at that time. I worked, and on some occasions I needed books from the University library, but I could not travel without police permission. I had to apply to the police for permission to go to Oslo to take out books from the library! I couldn’t do it by writing since I didn’t know exactly which volumes I needed. This was a bit cumbersome. The War did come to an end, however. I had been able to do some work in the meantime. I wrote some papers that I completed right after peace came in the spring of 1945, so I had a number of papers that I had finished and submitted in ’45 and ’46. In some ways, you see, even before the University was closed, the journals from outside had sort of dwindled down. First, things from abroad didn’t come, certainly not from the countries that were at war with Germany, and that included eventually the United States. After a while even the German publications started to come at more and more irregular intervals. And then they stopped. So there was really nothing. Even Zentralblatt stopped coming. In a sense you were operating then without really knowing what was being done elsewhere, in more or less complete isolation. It was not a bad thing, actually, because you were left to your own devices. You didn’t have that much distraction from periodicals that come in, and you feel that you have to look at them to see what is being done. Of course, you might worry that, my God, these things I’m doing might be somebody else’s, and it’s been done much better already. So I must say that after the War, when I heard that the Institute of Technology in Trondheim had gotten in things that hadn’t gotten to Oslo yet, I took a trip up to Trondheim to look through them. I was pleasantly surprised. It seemed that nobody had really duplicated the work that I had been doing at the same time. Before the War I had had the misfortune of doing a number of things where it turned out that somebody else had done them just slightly before.
The Institute for Advanced Study
MP: What prompted your invitation to the Institute in Princeton? Was it your dissertation, or was it the subsequent work that you did when you got out of the prison camp?
Selberg: I think it was Carl Ludwig Siegel who probably helped—you see, he was a good friend of Harald Bohr. He had contacted Bohr, who had been on the committee for my dissertation. By the time my disputation came, Bohr couldn’t be there because he was in Sweden. He had had to flee from Denmark. But he had been able to send his opinions, which were read by a Norwegian, Professor Thoralf Skolem. So I think it was Bohr who drew Siegel’s attention to me. I must say, though, that I had met Siegel earlier when he came through Norway in the early spring in 1940, before he left for the United States from Bergen—on the last ship that went out before the German occupation. He was very lucky. It was certainly Siegel who wrote me to ask me to apply to the Institute.
MP: It was also Bohr who wrote the citation in 1950 when you won the Fields Medal.
Selberg: Well, I always thought that if Bohr had not been the head of that Committee, I probably wouldn’t have gotten the Medal. I think these things matter. I don’t know. For some reason, the first mathematical congress I took part in as a member—I went to some lectures at the mathematical congress in 1936 [in Oslo], but I was not a member—it was a congress in Helsinki (or Helsingfors as you said in those days), a Scandinavian congress, and I gave a talk as did my two brothers, twenty-minute talks, of course. Actually I was surprised I was received in such a friendly way by some of these older mathematicians, not only Harald Bohr but also by Torsten Carleman, who had much less interest, I would think, in what I was doing. But he said some rather nice things about my work. I don’t think I was that impressive at the time. I was just barely 21. No, I think had Bohr not been on the Committee, it might not have been me. Of course, had Bohr not been on the Committee, it might not have been Laurent Schwartz either, because I know Bohr was very impressed by the theory of distributions. I think that on the whole, except in some rare case where there is someone really spectacular—Serre maybe, who would have gotten it no matter who was on the Committee—in other cases it’s a matter of taste. It’s the same with the Nobel Prizes. It’s partly a matter of chance. And in other cases it’s clear.
Figure 14.5 Selberg in his office in the Institute for Advanced Study (IAS) in 2005.
MP: Yes, absolutely clear. Pólya once gave me his list of the three nicest mathematicians he had ever known and Harald Bohr was on the list.
Selberg: He was an extremely nice man, and it was such a tragedy that he died so early.
MP: With all the applications of elliptic curves to problems in cryptography, it seems that suddenly number theory has become a branch of applied mathematics.
Figure 14.6 Selberg at the IAS. He was called a “mathematician’s mathematician” by Peter Sarnak of the IAS.
Selberg: It would have given great grief to Hardy!
MP: That’s right. Hardy would not have liked it at all. A colleague of mine, Paul Halmos, has said that applied mathematics is bad mathematics.
Selberg: Personally I see nothing wrong with mathematics being applied, as long as it’s good mathematics that is being applied.
MP: Of course, that is Paul’s point too. Did you know Hardy?
Selberg: No. I never got to England until 1971.
MP: And he died in ’47.
Selberg: I did meet Littlewood. He was rather old at that time, in his nineties.
MP: Yes, Littlewood visited here in the late 1950s. Of course, seeing Littlewood was fine, but seeing Hardy would have been interesting because he was surely a colorful figure.
Selberg: Of the two books that they wrote, Hardy’s A Mathematician’s Apology and Littlewood’s Mathematical Miscellany, Hardy’s book is by far the more interesting. There’s no doubt about that. Now I must say of Littlewood’s book, there are two editions, the original, and then there is one later where someone has added a lot of things. That was a mistake. It doesn’t add to the value. Much of the new material is rather trite, and though Littlewood may have said some of these things, they did not originate with him. Some of them Littlewood had heard already from other people. Littlewood’s book was interesting, but Hardy’s was a great piece of literature.
MP: I’m hesitant, though, to recommend Hardy’s book strongly to my students. When he gets off into the part about his being so proud that he’s never done anything that could be of use to humanity and so on, I wince a bit. I don’t know how my students read that.
Selberg: One must also wonder whether this might not have been written tongue in cheek, perhaps to irk some people. I think he liked to do that.
MP: I think so. I detect in his writing an inclination to provoke. He must have had an interesting sense of humor. So many of the Hardy stories are quirky and interesting.
Hardy Stories
Selberg: I heard a number of Hardy stories from Harald Bohr. Actually, once when I visited Bohr, the last time, in 1949, 1 was looking through a book that he had, where visitors had written something. Hardy had written about his experiences a good number of years earlier in the United States. He was in the habit of rating things.
MP: His famous lists.
Selberg: Of course, the Grand Canyon was rated very high. But there was another thing, in Princeton, that was rated high. It was a restaurant on Nassau Street that existed in those days. It was the first restaurant on Nassau Street that I ever ate a meal in, when I came to Princeton. It impressed me because the walls were all white tile, and it looked like a grand men’s room that had been changed into a restaurant!
MP: How attractive!
Selberg: He thought that had to be high on a list; it was just what Cambridge needed. It must have appealed to him for some reason. Actually Bohr had a number of stories about Hardy, partly related to his atheism. It wasn’t really atheism. His views were directed against God; they wouldn’t have any meaning unless God exists.
MP: There would be no adversary, no point to it!
Selberg: Here’s a story about Hardy: he was in Stockholm and went to visit the Cramérs. Harald Cramér’s wife showed him into the living room where Hardy observed a vase where there was a single rose and a book of poems by Rabindranath Tagore. It was open to a page with a poem to a rose. Marta Cramér waited for him to take in this carefully arranged thing and say something. So Hardy said: “Rabindranath Tagore, the bore of bores.” The perfect squelch.
MP: Something of a disappointment for Mrs. Cramér!
Selberg: Definitely.
MP: Apropos his views of Tagore, I found a letter from Hardy among Pólya’s papers that ranked poets. He gave one hundred points to Shakespeare, citing something in Macbeth, seventy-three to Milton, seventy-one to Shelley, thirty-nine to Tennyson, and twenty-seven to Browning. And he gave 0.02 to the American poet, Ella Wheeler Wilcox, who wrote “Laugh and the world laughs with you, weep and you weep alone.” He may have thought of her in the same terms as Tagore.
Selberg: My impression is, partly from that book he wrote, that he did have a good sense for poetry. And he had good taste.
MP: He was a cultivated man.
Selberg: Yes, and he had a good sense of language, much more so than most mathematicians.
Is Number Theory Too Hard Today?
MP: I have heard people say that number theory is just too hard now. The big problems have been around so long and have been looked at by so many good people, what remains is just too hard.
Selberg: One reason mathematics may seem very hard today is that, in a sense, too much is appearing all the time. It’s hard to keep up with it. I think it’s very essential to know what you should not read. It’s almost more important than knowing what you should read. It’s very difficult to say whether the old problems are the hardest. Actually, the oldest problem in number theory is the problem of the existence of odd perfect numbers, which has not attracted much attention. No really notable mathematicians have spent very much time on that. It may be partly because they consider the problem an unnatural one. But a lot of problems are unnatural—the Goldbach problem, for example. That’s certainly an unnatural problem. Why should you try to write even numbers as a sum of two primes? There is no earthly reason for it. Of course, that’s what it has going for it, according to Hardy; it certainly couldn’t have any applications! We are sometimes interested in problems just because they are hard! And sometimes if you succeed in doing one, you may have something that can be applied also to other things.
Figure 14.7 Selberg with a camera-shy tortoise in the Galápagos in 1997.
Still, there is something fascinating about number theory. I don’t think people will stop doing it, but it is definitely very hard. People tend to concentrate on the hard problems. They attract attention. The Riemann hypothesis is one. But at least it is a problem with meaning, not like the Goldbach problem. It’s a question you have to ask! There are a number of people working on it today, and some people think they are close to a solution. On the other hand, so have people thought several times in this century. I don’t know—they may still think so well into the next.
MP: Someone said once that number theory is that branch of mathematics everyone would work in if he or she were good enough.
Figure 14.8 John Nash with Selberg on the occasion of his ninetieth birthday celebration.
Selberg: I don’t think that is true. People have different kinds of talents and different types of intuition. There may be people who can do very fundamental work in topology and have a particular affinity for that and a special kind of intuition. They might not do very well in number theory or vice versa.
MP: But even some of the silly problems we talked about—the Goldbach for one—have an appeal early on to people because people can understand them. It gets them interested in mathematics, but then they realize that they have to earn a living and they go on and do something else. I often hear that people got interested in mathematics through some little number theory problem.
Selberg: Yes, that is true. I’m sure in past times many people may have gotten into number theory because of the Fermat equation. Gauss refused to work on it. But it did lead over the years to good mathematics—Kummer’s work, and, of course, the eventual solution, which came from completely different directions. It may be that way with the Riemann hypothesis as well; the solution may come through some connections that we have no idea about today.
MP: I recall that at the 1996 Seattle conference on the Riemann hypothesis, where you were the keynote speaker, there were physicists and people from a wide range of mathematical fields who were working on the problem.
Selberg: Sometimes a physicist’s way of thinking can introduce some useful ideas.
Winning a Fields Medal
MP: Let’s move back to your Fields Medal. I recall Lars Ahlfors saying that when he won it in 1936, he scarcely knew what it was. People hadn’t heard of it before. By 1950 people had heard of it, of course.
Selberg: I think it had largely been forgotten. I knew of it at the 1936 Congress in Oslo. There I actually met Ahlfors briefly, because he was a friend of my older brother. But I had forgotten about the Medal. So when I was approached before the Harvard Congress and told that I would get one of the Medals, it was a surprise. It was a very happy occasion because I didn’t expect anything. And for the same reason, there were many others who didn’t expect it. The problem with any prizes or medals is that they make one or two people happy, but they make a much larger number unhappy each time. So in that sense any kind of prize overall increases human unhappiness!
MP: Alas!
Selberg: So it’s questionable whether prizes are a good thing.
MP: Later, you won the Wolf Prize, and that involves real money.
Selberg: When I got the Fields Medal, there was some money with it. It was $1,000 Canadian. And at that time the Canadian dollar was roughly equal to the American dollar. Meanwhile, the value of the gold in the Medal has increased a lot. If it’s made of the same purity today as it was then, and I would assume that it is, those have some monetary value. It’s a very nice medal. But it’s not the kind of thing you can hang around your neck.
MP: You’ve done a number of things in mathematics that now carry your name, contributing to the mathematical language: Selberg sieves, the Selberg trace formula, the Selberg zeta function. . . . Does that give you great satisfaction?
Figure 14.9 Atle and Betty Selberg crossing a Norwegian fjord.
Selberg: Well, maybe, in a sense. Of course, as you know, in mathematics many things carry the wrong name. The Pell equation was not known to Pell. I recall that André Weil once said that if something is named for someone, in most cases that is not the person who discovered it. So we shouldn’t pay too much attention to these things. But it is nice to know that some of my work has not been wasted. It seems to have been useful to other mathematicians. And it has certainly been useful to me. It has given me a fairly good living
MP: It’s not very nice for people to have favorite children, but one might have a favorite mathematical result. Do you have a favorite theorem of your own?
Selberg: I have not really thought about that. I don’t think I really have a favorite result.
MP: Nothing of which you are most proud?
Selberg: No, I don’t really think so.
The Riemann Hypothesis
MP: Do you believe that the Riemann hypothesis is true?
Selberg: Yes, I do.
MP: At the 1974 Congress in Vancouver, I recall Hugh Montgomery’s saying in his talk that he believes it on even-numbered days and does not believe it on odd-numbered days or the other way around. You believe it on all days.
Selberg: Yes. At one time one might have had some doubts, but with the additional numerical evidence, I think there couldn’t really be any doubt. And I would say also that it would really be a blot on mathematics if a zero turned up off the critical line. Contrary to what Leibniz thought, this is probably not the best of all possible worlds. But with regard to the Riemann hypothesis, God should have gotten at least one thing right.
MP: So all the zeros are there.
Selberg: They’re all there.
The Institute—Anything Missing?
MP: The Institute at Princeton, it seems to me—and I suspect to most academics—has to be viewed as heaven on earth. Have you ever wondered about what you might have missed not working in a regular university with students and the variety of activities that go on in a more usual setting?
Selberg: There are two things about it. One thing is the missing students. The students are of various kinds. If you are in a university, one has many indifferent students, and one may have some good ones from time to time. And that is a great experience. I have missed something there. I have also missed something else: if you have to lecture regularly in a university you have to keep up with a number of things that you may tend to let lapse in your mathematical knowledge. Or you may refrain from learning some new things that you would have to learn if you were asked to give a course on it. So I may know less mathematics than I would have had I been a professor in a university. I would have had to keep up in areas that haven’t really interested me that much. I’ve had contact with bright young people at the Institute. Some people come there at a quite early age. But they already have direction when they come to the Institute. You don’t get the opportunity to shape their interests to the extent you can in a university. So I may have missed something there also. There is the fact too that if you are fifty years in the same place, you wonder whether it might not have been more valuable as a human experience if you had spent these fifty years at several different places, as most academics do. So I probably missed something, but I also gained something.
MP: While at the Institute you’ve done quite a bit of traveling too. You spent some time at the Tata Institute in Bombay, I recall.
Selberg: Well, many mathematicians do a lot of traveling. In some ways, I don’t really like traveling that much. I don’t like the process of getting from one place to another.
MP: Being there is fine; it’s the process of getting there that’s not fun.
Selberg: It’s not fun. It might have been more interesting in earlier times, going in a leisurely way by ship or train. But one has to make do with the life one has had. I cannot opt now for anything else.
MP: The other afternoon at the barbeque Brian [Conrey] and I were asked what our hobbies are. Both of us sat there and tried to think about our hobbies. I’m afraid that most of the things that I could think of—and I suspect the same may be true for Brian—are sort of attached to mathematics. I could think of collecting old mathematics books, and listening to music. What are your hobbies?
Selberg: Well, I do have some. In my youth I was very interested in botany, and I collected a large herbarium. I don’t collect plants anymore. But I’m still interested in them, and I probably know and can identify more species than most. I have collected seashells for quite a number of years.
Figure 14.10 Selberg on the Great Wall of China in 1998.
MP: But all those spirals—that’s mathematics.
Selberg: Yes, but some don’t involve spirals, the bivalves, for example. To a minor degree I’ve collected minerals.
MP: But those are often polyhedra.
Selberg: Yes, crystals are connected with group theory. That’s true. In general I’ve been very interested in nature, in flora and fauna. That has always fascinated me. I’ve not been interested in physics, in spite of its being closer to mathematics.
MP: I feel the same. I feel bad about it because I think I may be missing something, but. . . .
Selberg: I had some interest in rational mechanics when I was at the university. But that’s essentially just a branch of mathematics.
Mathematical Giants
MP: We came close to this earlier on when we were talking about Euler, that he was a giant in mathematics. Now let me ask another question: who are the giants of twentieth-century mathematics?
Selberg: I don’t think I know enough mathematics to say. The ones who have meant the most to me have perhaps not been the giants. Hecke among German mathematicians of this century would be one; in a sense I appreciated his work more than Edmund Landau’s, though I read a good bit of Landau. Hardy and Littlewood meant something; Ramanujan meant a lot. I think Ramanujan and Hecke influenced me most. I don’t think I’m competent to judge who is the greatest mathematician of this century. Hermann Weyl was a very impressive figure. He was very different from Carl Ludwig Siegel. A lot of Siegel’s work was the sort that when you saw it done, it still seemed sort of unbelievable afterwards. I think that Siegel’s lectures may often have had a negative effect on the audience in the sense that they saw all of this tremendous power being applied, but afterwards it still seemed as impossible as before. On the other hand, most of Hermann Weyl’s work was the kind where the problems seemed extremely hard, but once you have seen it done, it seemed rather easy and natural. These are probably the most important things in mathematics. In that sense, among mathematicians I have known, I would put Hermann Weyl at the top.
MP: You wrote somewhere about the distinction between two types of mathematicians, the problem solvers and the theory builders—of course, there’s a spectrum of people in between the extremes.
Selberg: You can undoubtedly make that distinction. In the long run, probably the people who introduce new ideas and concepts that lead to new theories and new directions may be more important than those who solve some problem. Of course, it can depend on what goes into the solution. It’s very hard to give a cut and dried answer to such things. In general, perhaps, it may be the things that lead to new theories, quite new ways of thinking, that are ultimately more important than solving specific problems.
MP: In one sense Andrew Wiles just solved a problem, but what he brought together to do it was extraordinarily ingenious and may have far-reaching implications.
Selberg: Oh, yes, that is true. There’s no easy answer.
