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I'm very honoured to have the opportunity to interview Roger his Brown and I'd like

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to start just by recalling a comment I made at your 60th birthday conference,

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actually, which is that I first heard about Roger Heath back when I was 15, and that's because I, I bought this book.

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It was my first mathematics book, Unsolved Problems in Number Theory by Richard Guy.

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And it's a it's a book with lots of beautiful problems in number theory.

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And a 15 year old can get something from it, even if they don't really understand all of the details behind it.

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And Roger is mentioned, I think, 25 times in that book.

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So as a 15 year old,

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I have the impression that the way mathematics worked was that Paul Eric posed problems and Roger Heath Brown solved that problem.

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So not much has changed since then about my impressions of Roger,

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but maybe we could go back then right at the beginning of your life and ask a few questions about your early life.

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So can you tell us a few things about where you grew up and how you developed an interest in mathematics?

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I was brought up in Welwyn Garden City. I went to local primary school, secondary school, grammar school.

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Um, my, my father was a research chemist, very scientifically literate, but not a mathematician as such.

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And I think my first real exciting exposure to mathematics came from looking at a book that my dad had problems by Duany.

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Oh, yes. And some of those have a distinct number theoretic nature.

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I remember one of them requiring one two. Factor II is the number 1111111.

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Which, of course, I couldn't do. Among various other things, I think something involving Fermat's Little Theorem.

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Yes. And.

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The book had some references, which as I grew older, I took up, I think one was maybe Oystein Orr's book.

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So this is when he was still at secondary school. I said, Yeah, yeah, yeah.

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And I chased up references that I had.

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A very good public library system. I got hold of Hardy and write various other books.

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And. Eastman's book on what was then called Modern Prime Number Theory, I think was in the 1930s.

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And did you have teachers at school that were encouraging this mostly?

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Well, I think, you know, by the time I was 16 or so, I was a number theorist.

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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

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or two problems that I think I asked my poor math teacher who explained Fermat's Little Theorem.

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But yeah, they would. They were keen to coach me in mathematics in general.

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Probably good for me because I was think too much number theory at times.

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So I mean, nowadays a student like you would almost certainly be involved in the Olympiad.

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So that would that don't come to you.

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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.

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And I think it was probably those sort of systems such as they were would have been restricted to independent studies.

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Somehow the impression I have, I think, or maybe the first time Britain sensitive to the Olympiad was 1967.

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So that would have scares you? Yeah, I went up to Cambridge in 69.

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Yeah. So it wasn't becoming a widespread thing until a bit later.

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Yeah. So from what you said it I mean, it sounds like a natural decision to study maths at university.

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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.

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And like me, some number of years later, you went to Trinity College?

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Yep. Is that. How did you hear about Trinity? As.

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As a place. My primus teacher just assumed if I was a good mathematician, I ought to go to Trinity.

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I had no say in it. Really. I did better now.

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So while you're at Trinity, I understand that. You had a supervision partner who subsequently got onto distinction in other areas.

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Can you tell us about that? So this must have been in my.

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Third year possibly. So my first supervision partner ended up I.

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Maybe he got a third, maybe in a special through same database?

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Possibly. I was rapidly moved on.

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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.

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And the story is that in part two it the Triforce.

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He was the person who was in second place by an off mask of anyone else.

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A decent to one was that person you?

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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.

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Don't, actually. Now that I've kept up with any of that.

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Yeah. Yes.

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And so well, after that undergraduate career, you made the decision to stay at Trinity and do a Ph.D. with Alan Baker.

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Well, because I did part three. Well, yes. Okay.

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That didn't seem to be much of a decision. It just happened automatically.

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I had no fallback position.

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I just somehow assumed that I would automatically go on to get a grant to do research and that they would take me in.

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Fortunately, they did. Yeah. And what made you decide to work with Alan in particular?

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Was he the person whose interests were just one of the Fields Medal at that time?

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I wasn't interested in his research. I wanted to make that clear to him.

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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.

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I mean, I had seen Hugh Montgomery's thesis, his lecture notes, multiplicative number theory, and that was clearly the thing for me.

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Did you see Mark O'Mara? He was a trustee as well.

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What did he overlap with you? No, he didn't. He must have.

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I never even met him until he came back later.

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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.

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Paddy Patterson possibly was the closest. Very rather different from what I was doing.

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It was very brave of Alan to take this, I think who was not going to work in the same areas.

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Yeah, well, I may return. I'm going to return in a little bit to actually the nature of that.

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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.

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Some of that can only be described as a gentleman mathematician going by the name of Cecil.

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John Alvin. Evelyn. Yes. He's listed his addresses, the St James's Club London or something of that variety.

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Could you tell us how this collaboration came to be? Yes.

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I guess this was during my first year as an undergraduate.

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My applied maths teacher from school noticed an advertisement in the Times saying A mathematical secretary wanted.

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I think it might have mentioned the Louisville function, which I had vaguely heard of.

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And so C.J. Evelyn had been a student of Hardy's has.

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One well known paper with with heart broken I believe that is the name does occasionally occur in the literature from the 1930s.

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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

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writing highly elementary papers about identities between arithmetic functions and wanted someone to help write them up for him.

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I see. Sounds like a dream job. Well, it paid well.

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Yes. I immediately realised that the mathematics was not at the level that I'd been hoping for,

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but he was paying me and he insisted that my name go on the papers.

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I suspect that it was only the fact that he donated sums of money or books, I believe,

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to the London Mathematical Society that persuaded them to publish his papers.

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But they were published, and that was when my first paper came out.

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And so to return to your thesis work and then I noticed looking at my son that I was starting in about 1978,

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basically a torrent of papers, issues for a rate of several a year.

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Did you hit the ground running in your first year? Were you suddenly proving results straight away?

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Yes, I think so. I mean, as I say, I got hold of Montgomery's lecture notes.

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As an undergraduate, I read particularly his stuff on zero density theorems and Alan Baker.

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Well, what does a supervisor do? You know, they see a theorem and they say, okay, generalise this algebraic number.

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So that was my first problem. And so I generalise this error density stuff to algebraic number fields and all that.

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Pretty straightforward really. And it got a little bit interesting where article functions, came out that little bit about that.

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But he didn't try and get you interested in transit. No, no, that's quite interesting.

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Okay. Strange because eventually I did some work that many, many years later that was based on.

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Yes, I was going to ask about grateful that I actually learned something this time.

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Well, actually, maybe we can talk about that now in case I forget. So this is, I guess, your work on how bombs expert.

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Yes, yes. So was that something that had those ideas been gestating since your thesis that scientists know so well?

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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.

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And every year the problem of higher problems, exponential, some would come right.

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And every year I would attempt and every year I would fail pretty much at the same point.

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And then. One year.

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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.

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P p is prime and the exponent is two pi times into the p over p squared.

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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.

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So both those factors make it difficult.

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And people had written about attempts to estimate this using base bounds where you get a compound is worse than trivial right now.

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Um. And then it must have been presumably sometime no early one Michaelmas term,

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relatively soon after problem session that Graham Everest came and was giving some

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lectures on the cave as a seminar on Marlow measures completely unrelated and.

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He mentioned the result of Deborah Volsky, which I remember Brian Burke and Baker talking about.

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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.

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And that made me think about the transcendence method, auxiliary polynomials.

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And suddenly the two came together and I realised that the problem that I'd wanted in connection with

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the honeybee on some might be attacked by means of the auxiliary polynomial method from transcendence.

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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

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you've thought about for ages and then something else totally different come out. Has that happened on other occasions?

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And that's must be the clearest example.

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I mean. Not not a problem that I've come back to again and again and again.

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So, you know, the problem on the covers problem Cubic Gal Summers, for example,

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was one where a method from what appeared to be a completely different area came along just at the right moment for me,

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and I was able to apply that to a cubic gal some was that was that with pattern that well every year well not

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every year but two or three times Patterson had given us an update on where he was on Cubic Gal sometimes,

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and so I was well aware of what his methods were capable of doing.

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And just at the right moment,

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I learned about the Vaughn identity and how it could be used for sums over primes which were not involving a multiplicative function,

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something which was specifically non multiplicative.

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I knew enough about Cubic Gotham to realise that they had this twisted multiplicative property, right?

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And that was enough to solve the problem, given quite a lot of help from Patterson on his stuff.

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And so to continue talking about somehow your earlier research career got a lot of famous results from that time.

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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.

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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%.

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So now I'm guessing that you're probably quite pleased with the first two of those.

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And think of the third as a not one of your.

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No, that's not one of my greatest. No, but I don't disown it in any way.

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I was going to ask, though, I mean, what do you think from that period up to about the mid 1990s,

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what what do you think of as your best guess pieces of work and.

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Possibly the 12th power moment for the Romans to function.

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Still something that I come back to every now and so maybe. Can you tell us a little bit about what those voters.

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One has estimates the size of the redundancy to function on the critical line.

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That height will be most of all what teach the one six or a little bit smaller and the £12 moment

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concerns the average of the 12th power of the Z to function and because it as far as we know,

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could be almost as large as t to the one sixth, then the 12th power could be as large as T squared near enough.

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And my results shows that this happens. And both one saw not much more.

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Of course. I mean, if we assume the Riemann hypothesis, then a much taller, much taller.

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But the in some senses this is the most powerful result we have on very large values of the Zeta function.

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And indeed it includes the point why is bound you can deduce the l infinity bound from this l 12 bound.

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So what is it about that result that that I like that you like the most?

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I mean, what techniques? Well, I suppose, first of all, it was very early on in my career before I.

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Before and certainly before I came to Oxford.

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It was a surprise element in it. That's always what one likes.

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And there were a number of useful corollaries results about the generalised divisor problem.

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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.

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And I should say that I mean, Heath Pride's identity is a it's quite a technical thing to state,

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but it was used, for example, by Zhang and his work on yes, famous recent work on small gaps of fame.

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So can you tell us about how that identity came about? Um.

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So Vaughan's identity was the big thing.

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At the time, I was completing my doctorate and.

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It was a beautiful way of handling sums over prime numbers.

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And it seemed to me that there were certain rather specialised situations where one might want to iterate it.

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And so I iterated it once, and there was one application where this gave a better result.

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And when you write down this iteration, it looks disgusting.

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But the so-called he's brown identity puts this on a much more straightforward footing and allows one to,

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in effect, iterate the vote identity as many times as wants.

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And you get a much better understanding of how the primes have to break up the the sum of a prime.

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But it is basically us special form of the binomial theorem doesn't see.

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I'd rather not looks at it that way. Of course, one has to think that such a thing should be out there, but.

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Yes. Yeah. And I, of course, I was familiar with which paper.

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So this was in the Canadian Journal Canadian Mathematical Society.

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It must have been. The late seventies and it was applied there.

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I feel the application was, I know a new proof of access on gaps between primes and yeah,

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it compared with the Vinogradov version, which in a sense is just as powerful.

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It's so much easier to use. Yeah, that's why everyone likes it.

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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.

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But then you moved in 1979 as a model? Yes.

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Can you explain how that came about? Yes. How that came. There were two factors.

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I just picked up with my girlfriend at the time. I felt it was good to move.

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And Brian Birch said there was a post coming up at all, so why didn't I apply?

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I find difficult to refuse any suggestion from Brian?

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I was probably under the false impression that if he suggested I should apply that I was almost certain to get the posts.

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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,

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particularly historically, but I gather that at that time the teaching load was quite substantial.

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Is that right? By today's standards, yes.

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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,

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which of course with preparation and marking is in like a full time job essentially.

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Plus 116 lecture course per year. So did this not impact on your research and then advice?

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Not in the slightest. Well, I just put up with my girlfriend.

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I didn't have a girlfriend equivalent. Spare their time.

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I lived in college. I didn't have to prepare any meals.

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Maudlin never seemed to have a full complement of students.

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I don't know that I ever really did 12 hours of teaching per week.

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It seems to me nowadays the best young mathematicians often get fellowships like the Society Fellowships or Episode C type scholarships,

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where they basically don't do any teaching until they're 30 or more.

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With these things are not so available in those days. I never heard of such a thing.

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Until such time as I thought these were for younger people and I was past.

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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.

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I had enough time for research. So, I mean, I've found in my career more than once that teaching,

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especially lecturing a graduate course, has actually been directly helpful to my research.

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Have you have you found that to be the case? Are you somewhat neutral on it?

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I'm pretty neutral, but I have one good example, which is just an example.

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I guess it was probably a problem that Brian Burke set for the elementary number theory course about

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Fermat's Last Theorem proved Fermat's the first case of Fermat's Last Theorem for Exponent five.

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This was long before Wiles, of course, and the first case of him because no doubt now I wouldn't know.

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So this is Fermat's Last Theorem in variables X plus Y to be ZB in variables which are not divisible by P, right.

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And there there's more than one elementary method to show that if the exponent P is five,

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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,

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but I presented the proof via Sophie Fisher Main Primes and this got me thinking about Sophie Germain Primes and I produced an argument.

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That would show that the first case of Fermat's Last Theorem was truth.

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Infinitely many prime exponents providing one hand some good information about the largest prime factor of p minus one.

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So I guess from my memory that has. So it wasn't known to be true for infinitely many exponents pillar that would

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have been the case had one known that there were many Sophie Germain primes.

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That's right. Which is a prime p famous prime open question.

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Yes. And so for you, my prime is a prime P for which two P plus one is also prime.

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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.

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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.

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And. I think Brian was going to a conference in New York about Fermat's Last Theorem.

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Probably connected with, um, with Elliptic Curves and decided to give a talk on, on my conditional result.

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Andrew Liska was in the audience. Some people knew about this because it never got published, but it was just a conditional result.

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And a few years later, uh, Len Edelman rediscovered a somewhat weaker version of this.

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Publicised it further. He got to hear about it with eventually able to prove the result about uh, approximate feature mean primes.

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And the result was a pair of papers. An invention is one.

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One is my joint paper with Len Eidelman on establishing the Criterion and Hoover's proving the result about.

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So this. This. I didn't know about this. So this together gives in so many, infinitely many cases of Fermat's Last Theorem.

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And this was the the machine is extremely weak but the strongest result no known

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and still presumably much easier for a layperson to but for an unknown number,

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there is certainly. Yes. Or does it require sort of deep and very severe theory and uh, the Bombardier Vinogradov Theorem, things of that.

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Good. No. I notice looking through your papers that it seems to me that around the mid 1990s your research took a somewhat

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different direction when you started getting interested in problems with more of an algebraic geometric flavour.

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Yes, and I would say that since then, more than half of your papers have been in that direction.

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So can you tell us a bit about how that direction started off?

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Well, I suppose I could think of it as being Brian Burton's influence, of course, but I suspect,

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in fact and I've always felt that I was influenced most of all by Davenport, even though I never met him.

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Yes, I was going to say he died before I went to Cambridge. And his papers on cubic forms were a strong influence.

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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.

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Which probably displays a distinct lack of knowledge about algebraic geometry.

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And over the last. 20, 30 years, I have slowly become slightly less ignorant of the subject.

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Well, it's relatively unusual for a more senior mathematician to kind of take on a completely new subject.

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Well, I think that other people have been more successful. This is my one attempt at learning some new mathematics.

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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.

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So within that sphere of work, which papers of yours stand out do you think I'm.

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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.

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And the. Are quite different papers. One is, I think, largely elegant, not entirely.

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And the other is foul. Which one is which?

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Well, the ten variable paper is large, elegant, but it has an awkward bit.

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The 14 very variable paper is foul from end to end.

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It's a good recommendation. You know,

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something else that seems to me to have changed around about the mid nineties

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is that prior to certainly prior to 1990 you'd only had I think one student,

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Graham. That's all that genealogy would admit to.

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And then subsequently you've had nearly 20 years.

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So how did that change? I think it's just.

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Fluke, really. I had other students and Graham Ringrose. He's the only one people have heard of.

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None of them wanted to. Almost none wanted to continue in academic academia.

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I had one student who left at the end of his first year, now had a couple who only ended up with me.

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So I know I'm not a good supervisor.

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Obviously, I'd wondered whether, because I've always had the feeling that in the eighties, somehow topology and geometry were king.

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I mean, anyone with that isn't the case. You do it. Maybe it's true.

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No, I had. Well, okay. Trevor Woolley applied to Oxford.

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We decided not to make him an offer he would have worked with.

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Okay. Yeah. I had.

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A very good student from Singapore who had a government scholarship which

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required him to go and work in the civil service in Singapore after graduating.

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And he never got back into into mathematics. And you're.

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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.

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No, I'm very pleased with the most recent ones.

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I mean. I suppose, starting with Tim Browning.

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Yes. Lillian Pierce, I suppose.

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Councillor says, you know, she was a student if not a doctoral student. Um, and then most obviously James Maynard.

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But I also think that phone Morrison has a feature to him.

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So yeah. Um, I think that, that.

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I thought maybe I'll just give them more encouragement and convince them that they should stay in academia.

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I mean, Chris Ringrose could have done, I think, but he.

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He wanted him to move off. Yes. So James Mignot is an interesting question.

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I mean, he. I'm trying to think how I might have reacted if I'd been his adviser.

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He might have come to me at some point and said, I want to try and prove that there are bank accounts between primes.

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I fear that I would have said the best thing that I might have said would be, Well, okay,

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think about that some of the time, but make sure you have some other problems on the go at the same time.

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What would be. Well and. I think he really started on this problem.

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So in his final year, if not partway into his final year.

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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.

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I think that there was already the germ of an idea worth exploring there,

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and one could see that due to some possibilities other people had looked at and therefore.

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Yeah, that it was worth spending some time on it. But.

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Obviously he didn't make the major breakthrough while he was with me at the time he had he left.

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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.

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Yeah. Okay.

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Well, I think coming towards the end of this. I just wanted to ask a few questions about.

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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.

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So I assume you're not planning to retire from doing research any time soon?

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No, I'm not. No, I feel like I'm slowing down.

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But yeah, it's so exciting. I want to be involved and hear what's going on.

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Give encouragement where one can. Are you prepared to share with us any specific research objectives or things that I've got?

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Three projects on the go at the moment in various stages.

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None of them seem to be earth shattering.

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So there's. Well, a problem that I've talked about a number of times already about gaps between primes.

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I've got a problem which involves the elementary class field theory primes for which 16 divides the quadratic class number four Q,

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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.

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Enough to keep me out of out of mischief for a while anyway. And some of us are of the belief that.

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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,

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which is 50 years old. You don't have any plans.

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Address that statement has. So I don't have any plans.

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If anything, I think a nice introduction or another nice introduction to analytic methods for diophantine problems would be nice.

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Partly because that's. Less written about and so many books.

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I agree that there's no perfect introduction to number theory, but there are lots of less than perfect auto centric.

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And I think apart from Tim Browning's book, there isn't really anything that quite.

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Covers the sort of material I'd like to see on an introduction to analytic methods for Diophantine equations.

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Now people have talked in the past about a book on Safe Methods, but I am such a perfectionist,

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so I wouldn't want to write a book about something where I couldn't give a an elegant, complete account.

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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.

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Okay. Well, finally, I mean, even now you don't have to teach and lectures on is more time in your life.

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Can you tell us about anything outside of mathematics that you're going to use this time?

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I feel I need more time for myself.

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I have a big project in. In field botany in Britain.

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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

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as it is up to the year 2020 and I've got 100 square kilometres of East Oxfordshire to look after.

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That's an area approximately the size of the convex hull of Oxford or something.

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Is that so? Yes. So that should be possibly more.

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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.

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Filled that a lot of mathematics to do. Okay.

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Well, thank you very much indeed. Good. Thank you.