I'm very honoured to have the opportunity to interview Roger his Brown and I'd like to start just by recalling a comment I made at your 60th birthday conference, actually, which is that I first heard about Roger Heath back when I was 15, and that's because I, I bought this book. It was my first mathematics book, Unsolved Problems in Number Theory by Richard Guy. And it's a it's a book with lots of beautiful problems in number theory.
And a 15 year old can get something from it, even if they don't really understand all of the details behind it. And Roger is mentioned, I think, 25 times in that book. So as a 15 year old, I have the impression that the way mathematics worked was that Paul Eric posed problems and Roger Heath Brown solved that problem. So not much has changed since then about my impressions of Roger, but maybe we could go back then right at the beginning of your life and ask a few questions about your early life.
So can you tell us a few things about where you grew up and how you developed an interest in mathematics? I was brought up in Welwyn Garden City. I went to local primary school, secondary school, grammar school. Um, my, my father was a research chemist, very scientifically literate, but not a mathematician as such. And I think my first real exciting exposure to mathematics came from looking at a book that my dad had problems by Duany.
Oh, yes. And some of those have a distinct number theoretic nature. I remember one of them requiring one two. Factor II is the number 1111111. Which, of course, I couldn't do. Among various other things, I think something involving Fermat's Little Theorem. Yes. And. The book had some references, which as I grew older, I took up, I think one was maybe Oystein Orr's book. So this is when he was still at secondary school. I said, Yeah, yeah, yeah. And I chased up references that I had.
A very good public library system. I got hold of Hardy and write various other books. And. Eastman's book on what was then called Modern Prime Number Theory, I think was in the 1930s. And did you have teachers at school that were encouraging this mostly? Well, I think, you know, by the time I was 16 or so, I was a number theorist.
They were encouraging me in mathematics in general. I remember my applied math teacher gave me access to all his past mathematical concepts and one or two problems that I think I asked my poor math teacher who explained Fermat's Little Theorem. But yeah, they would. They were keen to coach me in mathematics in general. Probably good for me because I was think too much number theory at times. So I mean, nowadays a student like you would almost certainly be involved in the Olympiad.
So that would that don't come to you. So I was a decent grammar school and we had a couple of people going to Oxbridge a year, but it wasn't one of the major players in the game. And I think it was probably those sort of systems such as they were would have been restricted to independent studies. Somehow the impression I have, I think, or maybe the first time Britain sensitive to the Olympiad was 1967. So that would have scares you? Yeah, I went up to Cambridge in 69.
Yeah. So it wasn't becoming a widespread thing until a bit later. Yeah. So from what you said it I mean, it sounds like a natural decision to study maths at university. Absolutely. I think, you know, when I was 13 or 14, I might have been a chemist like my father, but from then it was suddenly. And like me, some number of years later, you went to Trinity College? Yep. Is that. How did you hear about Trinity? As.
As a place. My primus teacher just assumed if I was a good mathematician, I ought to go to Trinity. I had no say in it. Really. I did better now. So while you're at Trinity, I understand that. You had a supervision partner who subsequently got onto distinction in other areas. Can you tell us about that? So this must have been in my. Third year possibly. So my first supervision partner ended up I. Maybe he got a third, maybe in a special through same database? Possibly. I was rapidly moved on.
So yes, in my second and third year, I was paired up with Singleton Lee, who is now or has been for many years, Prime Minister in Singapore. And the story is that in part two it the Triforce. He was the person who was in second place by an off mask of anyone else. A decent to one was that person you? I don't know. I may be even further down the list. I was in the same year as Bernie Silverman also did extremely well. Don't, actually. Now that I've kept up with any of that.
Yeah. Yes. And so well, after that undergraduate career, you made the decision to stay at Trinity and do a Ph.D. with Alan Baker. Well, because I did part three. Well, yes. Okay. That didn't seem to be much of a decision. It just happened automatically. I had no fallback position. I just somehow assumed that I would automatically go on to get a grant to do research and that they would take me in. Fortunately, they did. Yeah. And what made you decide to work with Alan in particular?
Was he the person whose interests were just one of the Fields Medal at that time? I wasn't interested in his research. I wanted to make that clear to him. When you asked whether I think it was pretty clear. Yes, he was happy to take me on on those times I wanted to do. I mean, I had seen Hugh Montgomery's thesis, his lecture notes, multiplicative number theory, and that was clearly the thing for me. Did you see Mark O'Mara? He was a trustee as well.
What did he overlap with you? No, he didn't. He must have. I never even met him until he came back later. He must have left a couple of years before me. Martin Hartley had left, and there was no one working in multiplicative number theory as such. Paddy Patterson possibly was the closest. Very rather different from what I was doing. It was very brave of Alan to take this, I think who was not going to work in the same areas.
Yeah, well, I may return. I'm going to return in a little bit to actually the nature of that. But I came across a fact which I didn't know before was researching this interview, which is that your first paper was with a gentleman. Some of that can only be described as a gentleman mathematician going by the name of Cecil. John Alvin. Evelyn. Yes. He's listed his addresses, the St James's Club London or something of that variety. Could you tell us how this collaboration came to be? Yes.
I guess this was during my first year as an undergraduate. My applied maths teacher from school noticed an advertisement in the Times saying A mathematical secretary wanted. I think it might have mentioned the Louisville function, which I had vaguely heard of. And so C.J. Evelyn had been a student of Hardy's has. One well known paper with with heart broken I believe that is the name does occasionally occur in the literature from the 1930s.
He was definitely a gentleman had gone on to manage the family estates and now in his sixties also came back to mathematics and was writing highly elementary papers about identities between arithmetic functions and wanted someone to help write them up for him. I see. Sounds like a dream job. Well, it paid well. Yes. I immediately realised that the mathematics was not at the level that I'd been hoping for, but he was paying me and he insisted that my name go on the papers.
I suspect that it was only the fact that he donated sums of money or books, I believe, to the London Mathematical Society that persuaded them to publish his papers. But they were published, and that was when my first paper came out. And so to return to your thesis work and then I noticed looking at my son that I was starting in about 1978, basically a torrent of papers, issues for a rate of several a year.
Did you hit the ground running in your first year? Were you suddenly proving results straight away? Yes, I think so. I mean, as I say, I got hold of Montgomery's lecture notes. As an undergraduate, I read particularly his stuff on zero density theorems and Alan Baker. Well, what does a supervisor do? You know, they see a theorem and they say, okay, generalise this algebraic number. So that was my first problem. And so I generalise this error density stuff to algebraic number fields and all that.
Pretty straightforward really. And it got a little bit interesting where article functions, came out that little bit about that. But he didn't try and get you interested in transit. No, no, that's quite interesting. Okay. Strange because eventually I did some work that many, many years later that was based on. Yes, I was going to ask about grateful that I actually learned something this time.
Well, actually, maybe we can talk about that now in case I forget. So this is, I guess, your work on how bombs expert. Yes, yes. So was that something that had those ideas been gestating since your thesis that scientists know so well? It's taking us many years on. You know, at the start of every academic year, Brian Birch would give a seminar on open problems. And every year the problem of higher problems, exponential, some would come right.
And every year I would attempt and every year I would fail pretty much at the same point. And then. One year. So maybe. Can you tell us what help runs exponential school? It's the exponential sum where the variable end goes from 1 to 2. P p is prime and the exponent is two pi times into the p over p squared. So that makes it, first of all, very high degree P rather than a constant degree and it looks like a some mod P squared. So both those factors make it difficult.
And people had written about attempts to estimate this using base bounds where you get a compound is worse than trivial right now. Um. And then it must have been presumably sometime no early one Michaelmas term, relatively soon after problem session that Graham Everest came and was giving some lectures on the cave as a seminar on Marlow measures completely unrelated and. He mentioned the result of Deborah Volsky, which I remember Brian Burke and Baker talking about.
And I thought it would be fun to go back and look at the proof and maybe improve on top of all its results, which of course I didn't manage to do. And that made me think about the transcendence method, auxiliary polynomials. And suddenly the two came together and I realised that the problem that I'd wanted in connection with the honeybee on some might be attacked by means of the auxiliary polynomial method from transcendence.
Because it's remarkable when this happens in mathematics. It sort of happened to me a few times as well when you sort of you've got some problem you've thought about for ages and then something else totally different come out. Has that happened on other occasions? And that's must be the clearest example. I mean. Not not a problem that I've come back to again and again and again.
So, you know, the problem on the covers problem Cubic Gal Summers, for example, was one where a method from what appeared to be a completely different area came along just at the right moment for me, and I was able to apply that to a cubic gal some was that was that with pattern that well every year well not every year but two or three times Patterson had given us an update on where he was on Cubic Gal sometimes, and so I was well aware of what his methods were capable of doing.
And just at the right moment, I learned about the Vaughn identity and how it could be used for sums over primes which were not involving a multiplicative function, something which was specifically non multiplicative. I knew enough about Cubic Gotham to realise that they had this twisted multiplicative property, right? And that was enough to solve the problem, given quite a lot of help from Patterson on his stuff.
And so to continue talking about somehow your earlier research career got a lot of famous results from that time. Let me mention a few that sort of a result, one results that lots of people know, like for example, the smallest prime in an arithmetic progression. What Q is that most Qs the 5.5 many twin primes if there's a single zero and then that Fermat's Last Theorem holds for quite 100%. So now I'm guessing that you're probably quite pleased with the first two of those.
And think of the third as a not one of your. No, that's not one of my greatest. No, but I don't disown it in any way. I was going to ask, though, I mean, what do you think from that period up to about the mid 1990s, what what do you think of as your best guess pieces of work and. Possibly the 12th power moment for the Romans to function. Still something that I come back to every now and so maybe. Can you tell us a little bit about what those voters.
One has estimates the size of the redundancy to function on the critical line. That height will be most of all what teach the one six or a little bit smaller and the £12 moment concerns the average of the 12th power of the Z to function and because it as far as we know, could be almost as large as t to the one sixth, then the 12th power could be as large as T squared near enough. And my results shows that this happens. And both one saw not much more.
Of course. I mean, if we assume the Riemann hypothesis, then a much taller, much taller. But the in some senses this is the most powerful result we have on very large values of the Zeta function. And indeed it includes the point why is bound you can deduce the l infinity bound from this l 12 bound. So what is it about that result that that I like that you like the most?
I mean, what techniques? Well, I suppose, first of all, it was very early on in my career before I. Before and certainly before I came to Oxford. It was a surprise element in it. That's always what one likes. And there were a number of useful corollaries results about the generalised divisor problem. For example, I'm. Now, something else that I guess must have come from that period is what's now known as Heath Brand's identity.
And I should say that I mean, Heath Pride's identity is a it's quite a technical thing to state, but it was used, for example, by Zhang and his work on yes, famous recent work on small gaps of fame. So can you tell us about how that identity came about? Um. So Vaughan's identity was the big thing. At the time, I was completing my doctorate and. It was a beautiful way of handling sums over prime numbers.
And it seemed to me that there were certain rather specialised situations where one might want to iterate it. And so I iterated it once, and there was one application where this gave a better result. And when you write down this iteration, it looks disgusting. But the so-called he's brown identity puts this on a much more straightforward footing and allows one to, in effect, iterate the vote identity as many times as wants.
And you get a much better understanding of how the primes have to break up the the sum of a prime. But it is basically us special form of the binomial theorem doesn't see. I'd rather not looks at it that way. Of course, one has to think that such a thing should be out there, but. Yes. Yeah. And I, of course, I was familiar with which paper. So this was in the Canadian Journal Canadian Mathematical Society. It must have been. The late seventies and it was applied there.
I feel the application was, I know a new proof of access on gaps between primes and yeah, it compared with the Vinogradov version, which in a sense is just as powerful. It's so much easier to use. Yeah, that's why everyone likes it. So after Ph.D., you took a research fellowship at home, so it was looking possibly as if you guys should just stay at Trinity forever. But then you moved in 1979 as a model? Yes. Can you explain how that came about? Yes. How that came. There were two factors.
I just picked up with my girlfriend at the time. I felt it was good to move. And Brian Birch said there was a post coming up at all, so why didn't I apply? I find difficult to refuse any suggestion from Brian? I was probably under the false impression that if he suggested I should apply that I was almost certain to get the posts.
But I was lucky and did. Yeah. And I assume that I mean, I don't know a huge amount about the teaching system at Oxford, particularly historically, but I gather that at that time the teaching load was quite substantial. Is that right? By today's standards, yes. It didn't seem substantial to me at the time for how much I was scheduled to do 12 hours of tutorial teaching per week for the eight weeks of term, which of course with preparation and marking is in like a full time job essentially.
Plus 116 lecture course per year. So did this not impact on your research and then advice? Not in the slightest. Well, I just put up with my girlfriend. I didn't have a girlfriend equivalent. Spare their time. I lived in college. I didn't have to prepare any meals. Maudlin never seemed to have a full complement of students. I don't know that I ever really did 12 hours of teaching per week.
It seems to me nowadays the best young mathematicians often get fellowships like the Society Fellowships or Episode C type scholarships, where they basically don't do any teaching until they're 30 or more. With these things are not so available in those days. I never heard of such a thing. Until such time as I thought these were for younger people and I was past.
I was no longer eligible. Almost to suppose I could have applied for a Royal Society professorship, for example, but I didn't feel the need for it. I had enough time for research. So, I mean, I've found in my career more than once that teaching, especially lecturing a graduate course, has actually been directly helpful to my research. Have you have you found that to be the case? Are you somewhat neutral on it? I'm pretty neutral, but I have one good example, which is just an example.
I guess it was probably a problem that Brian Burke set for the elementary number theory course about Fermat's Last Theorem proved Fermat's the first case of Fermat's Last Theorem for Exponent five. This was long before Wiles, of course, and the first case of him because no doubt now I wouldn't know. So this is Fermat's Last Theorem in variables X plus Y to be ZB in variables which are not divisible by P, right.
And there there's more than one elementary method to show that if the exponent P is five, then at least one of the variables must be a multiple of five. I probably didn't find the slick solution that Brian had in mind, but I presented the proof via Sophie Fisher Main Primes and this got me thinking about Sophie Germain Primes and I produced an argument. That would show that the first case of Fermat's Last Theorem was truth.
Infinitely many prime exponents providing one hand some good information about the largest prime factor of p minus one. So I guess from my memory that has. So it wasn't known to be true for infinitely many exponents pillar that would have been the case had one known that there were many Sophie Germain primes. That's right. Which is a prime p famous prime open question. Yes. And so for you, my prime is a prime P for which two P plus one is also prime.
That's right. And it's it's that way round. So you use the fact that 11 is also prime to handle the explosion equals five. Right. So that wasn't known. That was not known. And what I required was something weaker than that, which was just out of reach. And. I think Brian was going to a conference in New York about Fermat's Last Theorem. Probably connected with, um, with Elliptic Curves and decided to give a talk on, on my conditional result.
Andrew Liska was in the audience. Some people knew about this because it never got published, but it was just a conditional result. And a few years later, uh, Len Edelman rediscovered a somewhat weaker version of this. Publicised it further. He got to hear about it with eventually able to prove the result about uh, approximate feature mean primes. And the result was a pair of papers. An invention is one.
One is my joint paper with Len Eidelman on establishing the Criterion and Hoover's proving the result about. So this. This. I didn't know about this. So this together gives in so many, infinitely many cases of Fermat's Last Theorem. And this was the the machine is extremely weak but the strongest result no known and still presumably much easier for a layperson to but for an unknown number,
there is certainly. Yes. Or does it require sort of deep and very severe theory and uh, the Bombardier Vinogradov Theorem, things of that. Good. No. I notice looking through your papers that it seems to me that around the mid 1990s your research took a somewhat different direction when you started getting interested in problems with more of an algebraic geometric flavour. Yes, and I would say that since then, more than half of your papers have been in that direction.
So can you tell us a bit about how that direction started off? Well, I suppose I could think of it as being Brian Burton's influence, of course, but I suspect, in fact and I've always felt that I was influenced most of all by Davenport, even though I never met him. Yes, I was going to say he died before I went to Cambridge. And his papers on cubic forms were a strong influence.
I was trying to better these. And so I think the first paper in that vein was, was my paper on cubic forms and ten variables. Which probably displays a distinct lack of knowledge about algebraic geometry. And over the last. 20, 30 years, I have slowly become slightly less ignorant of the subject. Well, it's relatively unusual for a more senior mathematician to kind of take on a completely new subject.
Well, I think that other people have been more successful. This is my one attempt at learning some new mathematics. Yes. And it certainly convinced me that people in the past were held back by lack of appreciation of the geometry of some of these problems. So within that sphere of work, which papers of yours stand out do you think I'm. I guess the only two papers on cubic forms are the ones I'm most pleased with the one on ten variables, smooth forms and 14 variables in general.
And the. Are quite different papers. One is, I think, largely elegant, not entirely. And the other is foul. Which one is which? Well, the ten variable paper is large, elegant, but it has an awkward bit. The 14 very variable paper is foul from end to end. It's a good recommendation. You know, something else that seems to me to have changed around about the mid nineties is that prior to certainly prior to 1990 you'd only had I think one student, Graham. That's all that genealogy would admit to.
And then subsequently you've had nearly 20 years. So how did that change? I think it's just. Fluke, really. I had other students and Graham Ringrose. He's the only one people have heard of. None of them wanted to. Almost none wanted to continue in academic academia. I had one student who left at the end of his first year, now had a couple who only ended up with me. So I know I'm not a good supervisor.
Obviously, I'd wondered whether, because I've always had the feeling that in the eighties, somehow topology and geometry were king. I mean, anyone with that isn't the case. You do it. Maybe it's true. No, I had. Well, okay. Trevor Woolley applied to Oxford. We decided not to make him an offer he would have worked with. Okay. Yeah. I had. A very good student from Singapore who had a government scholarship which required him to go and work in the civil service in Singapore after graduating.
And he never got back into into mathematics. And you're. Well, let's see. I mean, it seems if your students seem to get better with time, I mean, you two really quite well students. No, I'm very pleased with the most recent ones. I mean. I suppose, starting with Tim Browning. Yes. Lillian Pierce, I suppose. Councillor says, you know, she was a student if not a doctoral student. Um, and then most obviously James Maynard. But I also think that phone Morrison has a feature to him.
So yeah. Um, I think that, that. I thought maybe I'll just give them more encouragement and convince them that they should stay in academia. I mean, Chris Ringrose could have done, I think, but he. He wanted him to move off. Yes. So James Mignot is an interesting question. I mean, he. I'm trying to think how I might have reacted if I'd been his adviser. He might have come to me at some point and said, I want to try and prove that there are bank accounts between primes.
I fear that I would have said the best thing that I might have said would be, Well, okay, think about that some of the time, but make sure you have some other problems on the go at the same time. What would be. Well and. I think he really started on this problem. So in his final year, if not partway into his final year. So he already had a lot under his belt and I felt it was quite safe for him to pursue a start moving on to something more high risk and.
I think that there was already the germ of an idea worth exploring there, and one could see that due to some possibilities other people had looked at and therefore. Yeah, that it was worth spending some time on it. But. Obviously he didn't make the major breakthrough while he was with me at the time he had he left. It didn't look like it was going anywhere much. Yes. And then well, the timing was in few weeks later, they'd done it. Yeah. Okay.
Well, I think coming towards the end of this. I just wanted to ask a few questions about. I mean, it seems to me that analytic number three now is in as good health as it's ever been in lots of great, absolutely brilliant young people. So I assume you're not planning to retire from doing research any time soon? No, I'm not. No, I feel like I'm slowing down. But yeah, it's so exciting. I want to be involved and hear what's going on.
Give encouragement where one can. Are you prepared to share with us any specific research objectives or things that I've got? Three projects on the go at the moment in various stages. None of them seem to be earth shattering. So there's. Well, a problem that I've talked about a number of times already about gaps between primes.
I've got a problem which involves the elementary class field theory primes for which 16 divides the quadratic class number four Q, square root minus P and a project that I've just been thinking about with Tim Trotter and the student has gone off to Australia. Enough to keep me out of out of mischief for a while anyway. And some of us are of the belief that.
The right sort of introduction to analytic number theory book has not been written that or at least no one's done better than Davenport's book, which is 50 years old. You don't have any plans. Address that statement has. So I don't have any plans. If anything, I think a nice introduction or another nice introduction to analytic methods for diophantine problems would be nice. Partly because that's. Less written about and so many books.
I agree that there's no perfect introduction to number theory, but there are lots of less than perfect auto centric. And I think apart from Tim Browning's book, there isn't really anything that quite. Covers the sort of material I'd like to see on an introduction to analytic methods for Diophantine equations.
Now people have talked in the past about a book on Safe Methods, but I am such a perfectionist, so I wouldn't want to write a book about something where I couldn't give a an elegant, complete account. But it may be that what's called for is precisely not a complete account. Well, short of incomplete, it sure is and could be very useful. Okay. Well, finally, I mean, even now you don't have to teach and lectures on is more time in your life.
Can you tell us about anything outside of mathematics that you're going to use this time? I feel I need more time for myself. I have a big project in. In field botany in Britain. I'm a keen, keen botanist. I'm going to be doing some recording for the Atlas 2020 project to record the floor of the British Isles as it is up to the year 2020 and I've got 100 square kilometres of East Oxfordshire to look after. That's an area approximately the size of the convex hull of Oxford or something.
Is that so? Yes. So that should be possibly more. Yes, that will keep me busy for for a while yet, but I'm going to be on my salary for four months next year. Filled that a lot of mathematics to do. Okay. Well, thank you very much indeed. Good. Thank you.