6th HLF – Laureate Lectures: The Riemann Hypothesis
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00:00
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Internet forumComputer animation
Transcript: English(auto-generated)
00:23
So, our next speaker also doesn't need much of an introduction. Sir Michael Atiyah, a mathematician known for his fundamental contributions to mathematics, namely the Atiyah -Zinger index theorem, and also his work on the intersection between mathematics and theoretical physics. He was awarded the Fields Medal in 1966 and the Adel Prize together with Isidor Zinger in 2004.
00:48
And today he will talk about his recent work on the Riemann hypothesis and in the email exchange before he promised to present it in a way that will also be accessible to the non-mathematical part of the audience.
01:04
Sir Michael, the floor is yours. Thank you Andreas. Well, first of all I should congratulate you and the HLF for being brilliant enough to combine John Hockcloth's talk with mine. I only realised listening to this talk, which I hope very much enjoyed, that his talk has about 50% overlap with my talk.
01:23
I leave that to the audience to find out. There's your test question. How can you see the correlation between his talk and my talk is actually very extensive? Okay, that's for a little problem to think about during my lecture. Secondly, I'm going to dedicate this lecture to my wife, who is seen there in her youth.
01:44
And she was a very serious woman mathematician long before there were other mathematicians. Now there's an exhibition here about women in mathematics. She took her PhD in 1954, long before, different era. And I want to dedicate this to her. She died, as you see, earlier this year.
02:06
Right, so now I've got to press the right buttons to this advanced technology. I'm older than John. When I was born, the computer wasn't even a glimmer in anybody's eye. Now I promised to talk about the Riemann hypothesis.
02:24
And so what is the Riemann hypothesis? First of all, who was Riemann? Well, he was the most famous German mathematician of his time. And he still is one of the great figures in history of mathematics. There is a picture of Riemann. You can recognize him because of his beard.
02:43
At that time, everybody had a beard. He lived a short life, you see, only he died at the age of 40. He did a fantastic amount. He collected works, occupied one volume. Every contribution in that volume is created in industry. If you want, you can be brilliant and write one work, and that's enough.
03:03
One work is a collection of ideas which has continued to the present time. Now the Riemann zeta function, which this is a conjecture about, is this function here. It's a function of zeta s. It's the sum of the reciprocals of the integers to the power s.
03:21
And if you think of s not just as a real number, but as a complex number, Riemann showed that it was an analytic function of this complex variable. And the Riemann hypothesis is very simple. Very simple. What does it say? This Riemann hypothesis was formatted in 1859. The computer wasn't even dreamt of, of course.
03:44
And it says that in the critical strip, the critical strip is when x slides between 0 and 1, any 0 of this function has to be on the line x equals a half. In this critical strip, in the middle, there is a line x equals a half, and all the zeros of this function should be on that critical strip.
04:03
And this has been verified numerically for millions and millions. Every computer you can throw at it always finds the zeros on this line, but there is no proof. Computers can't prove, can't produce proofs yet. And so the question is, how can you prove this?
04:20
And that's what I claim to show you, a proof. And so that is Bernhard Riemann. That's starting it off. Now, I'll go backwards in time, because Riemann was not the first man to think of this function. An earlier man, another great mathematician, greatest mathematician of the 18th century, was Leonhard Euler.
04:41
Euler, of course, was a famous Swiss mathematician, and he observed very fundamentally the formula called the Euler product formula. You can write this zeta function not only as an infinite sum, but with integers, but as an infinite product over the primes. If you work out this formula, what does it say?
05:02
It basically tells you that the sum is equal to the product because every integer has a unique factorization in terms of primes. We all know that six is two times three, and this formula embodies that. So this is a theorem that relates infinite sums, infinite products, and with that formula, Euler was able to do marvelous things.
05:24
Euler became very famous as a very young man. So young men in the audience, think, if you want to be famous, what do you do? Well, you solve a very difficult problem early on. The problem was, if you sum this series, one, two, three, four, what happens if s equals two?
05:42
What is the value of the zeta function at two? Well, it's easy to see that some number is a convergent series. It adds up to a number, but what number? All the whole mathematicians in Europe couldn't solve this problem. Euler, as a young man, found the formula. What was his formula? It is the most beautiful formula.
06:03
It wasn't any old nasty number. The value of zeta function at two was pi squared over six. You all know what pi is, I hope, but if you don't know what pi is, you can read about it if you go back and read Archimedes. But pi squared over six was a magical number. Everybody was staggered.
06:21
He became immediately famous. He was appointed to the emperors, Catherine of Russia and Frederick of Prussia. They all offered him jobs. He was like, you know, Messi or Ronaldo in football now. Everybody wanted him on his team and they paid him lots of money. He had a good life.
06:41
And now he wrote, his collective work occupied 73 volumes. They're still publishing them. They're still unpublished volumes. And these 73 volumes are fantastic. But you see, Euler published one volume. Euler published 73. Well, they're both great mathematicians, so you can be great mathematicians in many ways.
07:03
You can publish one volume or 73. But you have to choose the right volume, not any more volume. Now, why is the Lehman hypothesis interesting? The hypothesis says some function has its zeros on this line, but why is it interesting?
07:22
And also, why is it difficult? We're told it hasn't been solved yet, 170 years afterwards. Well, the reason is, because of the connection with the primes, the prime numbers, you know, look at prime numbers. They go two, you don't count one, two, three, four is on a prime.
07:42
Five, yes, six is on a prime. Seven, yes, eight is on a prime. So you see, they're irregular. And as you go further and further out, they seem to have very odd behavior. So very local irregularity, but it turns out that for the very large numbers, they have an asymptotic regularity.
08:01
So they have both asymptotic regularity and the local irregularity. And it's very intriguing. And people have been struggling with this for hundreds of years, and still are. If you want to work on a problem, number theory has infinitely many problems. Now, for example, you ask the following question, very naively.
08:21
How many primes are there less than n? n is a big number. You choose n, whatever you like. It's a nor notoriously difficult question. You get approximate answers, numbers depending on n, and it is a fundamental problem. How do you know how many primes there are less than n? It turns out the Riemann hypothesis is interesting,
08:42
because if you can prove the Riemann hypothesis, it leads to the best possible answers. We know what the best possible answer is. The Riemann hypothesis tells us. And if you don't get the Riemann hypothesis, you don't get the best answer. So it's very simple. Solve the Riemann hypothesis, and you become famous. Well, if you're famous already, you become infamous.
09:07
Now, let me give you a short history of prime numbers. Prime numbers have been interesting for mathematicians ever since there was any mathematician. Euclid knew a lot about primes. He proved there were infinitely many primes. And, of course, prime numbers were studied before Euclid.
09:23
And he had the smallest of fellow mathematicians who studied primes. Gauss, Euler, Riemann, more recently Hadamard, Seiberg, Banglans. Every mathematician worth his salt, or her salt, was interested in primes.
09:41
And here are these pictures below. I can't show you as many pictures as my friends in computer science who can conjure up millions and millions. I only have a few. So these are two pictures here. The picture of the man, he was called Ramanujan. He was the most famous, still is, one of the famous and romantic figures in history of mathematics.
10:06
He was self-taught genius. You see, he had a short life. He died quite young. And his story is actually made into a film. And on the right-hand side, you see a bit from the film.
10:20
The film was a very good film. It was called The Man Who Knew Infinity. And it won Oscars for the leading actors. And it's a very beautiful film. I recommend the film for you. You may not know anything about the prime numbers, but you'll enjoy the film. It's a very good movie. Now, this picture on the right is a snippet from the film.
10:40
And it shows you two characters. The man, the tall man, who was Indian, he's acting Ramanujan. He's a very good actor, Dev Patel. Unfortunately, he's twice as big as Ramanujan was. So it's a bit... But okay, actors can't always be the right size. And the man next to him is a very famous British mathematician.
11:04
Well, he's actually Jeremy Irons, who got the Oscar. But the person he's playing was called G.H. Hardy, who was one of the famous mathematicians in Cambridge, in the world, in his time. And in the corner of the...
11:21
In the corner... Let's see, there. Just there. That is the new room in which Isaac Newton lived. So this is a college which has a mathematical tradition, going back at least to Isaac Newton. And this picture is a very interesting picture. I leave you with a little question. For those of you who like questions, I don't know if there's a prize.
11:43
And the question is, you can see from the picture, there was a lot of sunshine. Obviously, the sunshine. Why is the man in the middle, was an umbrella? Think about that. Come and tell me afterwards what your project theory is. Why he got an umbrella, obviously, nice and sunny.
12:01
Well, the answer is very interesting. But, I mean, think about it during the lecture, because my lectures are going to be rather boring. So now... Now, among the theorems that you can prove with the Riemann hypothesis, and tell you in particular there is an asymptotic regularity of a certain kind,
12:23
but actually that can be proved separately without the Riemann hypothesis, and it's called the prime number theorem. Now, the prime number theorem gives you some information about asymptotic distribution of primes. How many... As you go very far out, how many primes do you find asymptotically?
12:41
And there's a very precise formula, and there's beautiful analytical proof of this by two famous French mathematicians, and their pictures are down below. Hadamard and de la Vallee Poussin, a rather nice name that rose off the tongue. And you see they dressed a bit differently.
13:01
De la Vallee Poussin, obviously old style, high collar, very handsome looking man. And then on the right hand side, turned out there was an elementary proof, which didn't use any analysis. People thought you couldn't give an elementary proof, and that proof was given by Aspie Selberg, a famous Norwegian mathematician,
13:23
who was my colleague at the Institute for the Republic of Saudi and Princeton, and there's a nice picture of him. And there's one theorem which was true at the time, in the 90s, when I was a young professor in the 60s, and the theorem was that if you proved the prime number theorem, you were immortal.
13:41
And there was no counterexample. See, they lived to be nearly 100, but then sadly, eventually, somebody died. But for a long time, it was true that if you proved the prime number theorem, you were immortal. Immortal in the real sense, not just in a historical metaphorical sense.
14:00
So don't forget, if you prove really good theorems, you become immortal. You are one of the immortals. Well, we all search for immortality. Where you get real immortality is improving good mathematical theorem. Now, this story has got to be a lot of sidetracks and backtracks.
14:24
I'm going to talk about quaternions. Now, I'll tell you later on what the quaternions got to do with prime numbers and the Riemann hypothesis. It's totally irrelevant, you think. And not so. And what are quaternions? Well, you know what real numbers are.
14:43
Then you have complex numbers, X plus IY. Then if you go on and add a J and a K, you get quaternions. And the formulas that I, J, and K have to satisfy, well, I squared is minus one, and so J squared is minus one. But the important thing is that in addition to that, multiplication of two numbers depends on the order.
15:08
JK is not the same thing as KJ. And this was the first time anybody thought about the idea that multiplication was not cumulative. This was the great discovery of the man Hamilton.
15:21
Hamilton was a great mathematical physicist, a genius. He was appointed Astronomer Royal of Ireland before he graduated. He was still 21. He hadn't taken his final examinations, but he was made Astronomer Royal of Ireland.
15:40
His name is known to all physicists, theory Hamiltonians. He was really one of the giants of physics in his era. And these pictures show you the day he discovered quaternions. Now, this is a really historical fact. Many discoveries are apocryphal.
16:01
You know, it's true that Archimedes jumped out of his bath, shouting Eureka, running down the streets naked, when he discovered some facts about what happens when he's in the bath. And even more famously, Isaac Newton is meant to have discovered the theory of gravity when an apple fell on his head. Bang!
16:20
The apple tree still exists, I've seen it. Well, it's in attendance. But these are apocryphal. There's no evidence. There's no eyewitness telling Archimedes ran down the street naked. There are no real witnesses to Newton and the apple. But Hamilton is a witness, because he had this brilliant idea about quaternions
16:44
as he was crossing a bridge in Dublin. He was Irish, lived in Dublin. And crossing this bridge one day, 1843, a flash of inspiration struck him, and he suddenly realized the quaternions. And if you want to know what the inspiration is, you look, there's the bridge, there's the plaque, here's a copy of the plaque.
17:05
You go to Dublin, you're asked to see the bridge where Hamilton crossed the bridge. There are other famous bridges where Caesar crossed the bridge. But Hamilton's bridge is historical, really happened. And it's inscribed there, and actually it's a great event.
17:20
So this is one that is really recorded in history. Okay. And you've learned something then, you may not have known that. And it seems to have nothing to do with Reman, but okay. We'll come to that. Let's see. I've got slides, but sometimes when you press a slide, you get two for one. I've got...
17:41
No, that's right. Now, let's go back to Euler. Euler and complex numbers. The most beautiful formula in mathematics. If you ask any mathematicians, almost anywhere in the world, what is the most beautiful formula in mathematics? They'll say it's this one. E to the pi i equals minus one.
18:02
It has all the important numbers in mathematics. Pi, which was really discovered by Archimedes. E, which was invented by Euler. And the square root of minus one, which is invented by many people. And there's one formula which puts them all together in a small package. It is the most succinct, beautiful formula in all of mathematics.
18:22
I think everybody would agree. I try to explain this to non-mathematicians. It is the mathematical equivalent of the short phrase in Shakespeare's Hamlet, saying, to be or not to be. It is very short, but very deep. And this is the mathematical equivalent of Hamlet's statement.
18:41
It is something to ponder over. So, it's a beautiful formula. Now, when we go beyond, he involves real numbers and complex numbers, because i appears. Archimedes didn't know about complex numbers. He couldn't do it. Euler did. Euler did it.
19:00
But now we've got quaternions. So what's more natural than to find a quaternionic analogue? Okay, I can give you a problem in the back. What is the quaternionic analogue of Euler's formula? So, what is it? Well, it turns out that it's there. But you've got to look for it. It turns out that you have to go back in history, not to Euler, but a little bit later,
19:25
to two great mathematicians of our... Well, sorry, I'm a 20th century man of the 20th century. But that's about 70 years old or more. The two mathematicians involved are Johnny von Neumann, who was undoubtedly a genius.
19:47
And here is a picture of von Neumann. He died in 1855. I met him briefly before he died. And he, well, von Neumann did everything. He was one of the pioneers of computers. He developed weather forecasting.
20:01
He did work on foundations of quantum mechanics. He developed logic. Almost everything. Everything you can think of. Except one thing he didn't do was number theory. Everything else, he did. And he was a real genius. In my sort of... I use the word genius very rarely.
20:21
And who are the three geniuses I recognize? First is Wolfgang Amadeus Mozart. We're here in... Mozart was somewhere not very far away. He was undoubtedly a genius. Secondly, there is von Neumann and Ramanujan. These are undoubtedly geniuses. And beyond that, I use the word very sparingly.
20:42
I don't recognize any old chap as a genius. So von Neumann, Ramanujan, Mozart are my trio of geniuses. Is geniuses the plural of genius or is it genii? I'm not sure. Genii comes out of a bottle. It's different.
21:01
Now, if you put together the work of von Neumann... Now, sorry, I didn't mention Fritz Hitzelbrock. Fritz Hitzelbrock was really in Germany. He was the most famous German mathematician of the post-war era. He did everything in Germany. He helped unite East-West German mathematics. He was my friend, my collaborator.
21:21
But he was a great man in his time. And he died some years ago. And this is in some sense a tribute to him also. And if you combine the work... Now, von Neumann and Hitzelbrock together, and that was what I did. All I did was the middleman.
21:42
I took one genius, another genius, put them together, and you get three geniuses instead of two. No conservation law for geniuses. So if you do that together, then you end up with a formula that is a good generalization of Euler's formula. To the Quaternions. And what does it look like?
22:01
Well, I gave it a name. And obviously what it has to be called, it's called the Euler-Hamilton formula, because it uses the work of Hamilton on Quaternions. And the formula is there. You see, this is where things go wrong.
22:27
I think we're right. Here we are. Euler-Hamilton formula. E to the... Now, this is a letter which we've got used to using the Latin and Greek alphabets.
22:40
No. Alpha beta gamma is Greek. ABC is Latin. But there are other alphabets to be used. And so here, in honor of the great Russian mathematicians and others, I thought we'd use the Cyrillic letters. And so this here, this funny letter there, if you don't know it, some of you, many of you will.
23:05
This is ж. It's a Russian letter. It's the first name of people like Zhdanov Zhukov, and it's a well-known Russian letter. So I use that. It's a nice letter because it doesn't... You're not likely to confuse it with anything else.
23:20
I looked through the Cyrillic script to find one, and it was the only one that suited the purpose. So I borrowed it. Well, I kept it out, stolen it. And the formula says... Well, it looks the same, doesn't it? It says... What's happened here?
23:40
Yeah, there we are. и equals minus one. And if you compare that with that one, you see that there's a little formula that relates pi and ж and I. Okay. And that's a good formula. Why is that a good formula? Well, I'll explain in a short while why it's a good formula.
24:01
The reason is when you do something, you want it to be perfectly natural. And the difficulty is if you write J or you've chosen and made a choice, you don't want to make a choice. You want to do things beautifully. Or another way of saying it is you want it to be... Well, let me put it this way. The quaternions are not cumulative.
24:23
So multiplication on the left, multiplication on the right, not the same thing. And so you want somehow to differentiate that. And this is what this function does. Now, let me go on. Yes. It turns out that the distinction between left and right multiplication is actually something you can measure.
24:41
It's called conjugation. You multiply on the left and the right. And if they're commutative, you get nothing. You get one. It's not commutative. You get something. So this is what is called, by his book, used the name Todd to this function. He discovered it. And Todd was my teacher.
25:00
And his book worked in the area. And he gave the name Todd, Todd polynomials, to this discovery. And so I'm copying... I'm using the same terminology. Why? Use another letter when Todd is already for you there. Now this function, this Todd function, is very well worth studying. So I can spend a bit of time on it.
25:21
This function is a very remarkable function. Let me give you its properties. Well, I've written down here, the first line tells you it's, I call it a weekly analytic L2 function. Those of you who are not mathematically trained may not know what those words mean. But don't worry, if you're a mathematician, you know what every word means there.
25:42
Analytic, weak, L2. All these are standard terms of terminology. And it's defined for a complex variable which is not zero. Now, you may think, well, what does weak mean? And at the bottom, I've got the properties. Weak means that it's true on compact sets.
26:03
If something is true on every compact set, you say it's weakly true. It converges weakly. That's what weak means. Analytic, everybody knows, analytic means it has a power series expansion. And it's a beautiful function. All our nice functions are analytic functions,
26:20
polynomials, exponents. So analytic functions are the basis of mathematics. L2 means square interval, means you can integrate them in the L2 norm. And you put all these function properties together, this functions all these properties. Now, if you're a good mathematician,
26:41
or even only half a good mathematician, you will say such functions don't exist because, you see, it follows from this that such a function can have a compact support. Now, an analytic function can't have a compact support. I mean, that doesn't make sense. But weakly analytic functions can.
27:00
So you see, the words here are very carefully chosen. So the properties I've written down at the bottom are it's analytic on compact sets. So any compact set is given by an analytic function. If the compact set is convex, it's actually a polynomial. So you see polynomials popping up.
27:21
And you remember the importance of polynomials was in John Hopcroft's talk. This is the importance. They are polynomials, but they're not always polynomials. They start off as being analytic functions and they can have compact supports while a polynomial doesn't usually have compact supports. So this is a very, very clever function. And now it's how it's defined.
27:40
You might say, you can give a definition, but where on earth do you find these animals? What zoo? What desert do you have to explore to discover such a fine, odd beast? Well, actually, it turns out it's defined by an infinite iteration of exponentials. We know how to form infinite sums. We've been doing that since Euler.
28:00
We know how to form infinite products. We've been doing that since Euler. We don't know how to form infinite exponentials. Infinite exponentials is a number like this. Two to the power of two to the power of two to the power of... And keep on going to infinity. What do you get? Well, you get a fantastically big number. After a while, it exceeds the size of any computer you ever imagined.
28:21
And a little bit longer, and it leaves the last computer way in the background. It is ultimately enormous. Actually, of course, it's infinite. But people aren't afraid of infinities. Particularly, they can calculate the ratio of two infinities and get a finite number. So every infinity can be measured in some clever way
28:43
comparative to other infinities. So if you're a mathematician, you're not scared by this. Nevertheless, nobody has really ever thought seriously about infinite iteration of exponentials. It's a dangerous game. In fact, you might think it's dangerous territory to enter into this game
29:01
and you'd be wise to keep away from it. Except that, I don't need to do that because this was done by von Neumann. And von Neumann's such a genius. If I have him on my side, it's like going into a room with the best lawyer you can get. If you've got the best lawyer, nobody can argue with it. Nobody would dare to argue with von Neumann about logic and things like that.
29:22
So von Neumann tells me it's okay, and I just take his word for it. I can see what he's saying, but I wouldn't believe myself. I believe von Neumann. And I think von Neumann is well-known, of course, to all people in computer science, as well as mathematicians. He was really guru. So with von Neumann behind me, this is all great, marvelous stuff.
29:45
These things exist. And the infinite iteration of exponentials is actually a very powerful technique because when you do things infinitely often, they get more and more powerful all the way through.
30:00
Now, okay, so now this Todd function is really a magnificent animal. It does exist. It was discovered not in the steps of the desert, but by going back and searching the literature and reading von Neumann's papers. It was there. All he had to do was find it. Now, okay, that was all very nice.
30:22
One on earth has got this to do with the Neumann hypothesis. Now, one thing you can tell, if somebody wants a claim to prove the Neumann hypothesis, there have been so many false proofs. If you press Google, you will get 100 false proofs of the Neumann hypothesis straight away. Well, there are proofs which are claimed, not believed, disproved.
30:43
You know, nobody believes any proof of the Neumann hypothesis because it's so difficult, nobody has proved it, and so why should anybody prove it? Now, unless, of course, you have a totally new idea. If somebody comes along and says, I'm just using the old ideas cleverly, they'll say, now, that doesn't work. There are cleverer guys than you who've tried to do this, it doesn't work.
31:03
If you say, I've got a totally new idea, you say, oh, well, that's interesting. Let's see what you've got. Okay, so I claim I've got a totally new idea. I've told you what a totally new idea is. It's new to von Neumann. So, that is a powerful method. Now, using this Todd function, and now the Todd function, I studied that.
31:27
See, people prove difficult theorems, for example, sometimes by spending 10 years of their lives, single-minded purpose, like Andrew Wiles proved Fermat's Last Theorem very famously. I wasn't trying to prove the Neumann hypothesis. I hardly knew what the Neumann hypothesis was.
31:41
Not the number theories. I knew this was way outside my area. I was doing them quite different. What was I doing? I was doing something quite different. I was discussing a problem in physics. It's called something called the fine-structure constant, called alpha. And here is, I identify this alpha with that number, or inverse,
32:03
and its well-known number is 137.035999. So, it's a well-known number like pi. Now, it turns out that the Neumann hypothesis was just a bonus. I wasn't looking for it, and I was trying to prove this. Let me digress now and tell you why the fine-structure constant was interesting.
32:22
Why is the fine-structure constant, what is it, and why is it interesting? Well, I will quote you to two people. One is Richard Feynman, a very famous theoretical physicist, Nobel Prize winner, and a man who can express himself with great force. The other one was someone not so well known in the mathematical world called Jack Good.
32:43
He was actually a statistician, cryptologist, he worked with Turing during the war, and he went to America later on, and he is actually quite well known, I think better known to scientists than he is to mathematicians. So, these are two very famous people, and Good didn't die that far back.
33:01
Now, here is what Feynman said. I'll read it to you because it's worth saying. Alpha is this number, and what Feynman says is, where does alpha come from? Is it related to pi, or perhaps e? It's a mathematical number, nobody knows. It's one of the great damn mysteries, sorry for that, I'm not swearing, he is swearing.
33:23
It's like the great damn mysteries of physics, a number that comes to us with no understanding by man. Sorry, man, woman. You might say the hand of God wrote the number, and we do not know how he, with a capital H, pushed his pencil. So, you see, he is very, very sort of theological.
33:42
What is this number on earth? And Good was more modest. He just said, we need a fundamental theory that provides a platonic explanation of alpha. We don't want just numbers. Somebody can, lots of people try to find what alpha was by playing around with formulas involving pi, and you can make almost any number you like.
34:02
Arbitrary accuracy by fiddling around with a few formulas. That's called numerology. That would be no good for good. If you mean like good, you have to be good. So, I claim that my explanation up there satisfies those. So, I worked for a long time, well, not so long.
34:22
I went to a conference in Singapore, I think in January last year, and I met a lot of good physicists, Nobel Prize winners, and particularly Gerard Tuft, and they convinced me that understanding the fine structure content was really a fundamental question.
34:41
So, but it was a mathematical question. Here's a chance for a mathematician to contribute to theoretical physics. So, I struggled, and eventually I found the answer, and the answer was what I've given you. It came from this combination of ideas, going back to von Neumann and his book. So, that was the fine structure, and I was very pleased.
35:02
I put my feet up, I opened my champagne bottle, metaphorically, but if you want to buy me some champagne, I'll be happy. And then suddenly one day, you know, I started to look down, and below me, I sort of found this arrow proof hypothesis.
35:23
I stumbled over it. I wasn't looking for it. I wasn't trying for it. I was solving a quite different problem. This is a good example for a young girl at the back. Sometimes the problems you get are by indirect approach. Solve problem in A, if it's hard enough, you might solve problem in B, which you don't even know about.
35:40
That is the beauty of mathematics, the transposition of ideas from one front to another front, you know, across centuries or across oceans. So, this is an example. Now, sorry, I should now get back. How does this help to prove the real hypothesis? I told you all the ingredients. Where's the punch line?
36:02
Here is the proof. This is the entire proof on one page. So, we're going to prove the real hypothesis by contradiction. The real hypothesis says that in the strip, and there is a picture of the strip. Why is this not working? I'm missing this.
36:31
Ah, there we are. There is the strip. Zero one, critical lines of the half, and you assume there is a zero not on the strip.
36:40
You assume that this yellow cross is the zero that shouldn't be there. And the claim is, you try to assume it's there, goes through an argument, and the argument leads to a contradiction, therefore the assumption was wrong, therefore that yellow cross wasn't there, therefore the theorem is true. That's the proof by contradiction and the standard proof. Most proofs are of that kind.
37:02
So, you go ahead. Now, I can look at the black picture here. So, what do you do? Now, you use this function t. Now, this function t is a nice function, in particular, it composes one of these weak analytic functions, you compose one with another one, you get another one, there well behaves. So, you can take your function t,
37:21
you compose it with the zeta function, which is analytic, and you look at this only in a compact set, where you keep away from the bad points of the zeta function. In that set, this is defined, and I made a small change of variable, so that the yellow cross becomes the origin. Okay, very small change of variable,
37:42
and notice that the rectangle that you draw in the critical strip, which is slightly away from the edges, is a convex set. And so, by the time you change this, you change the yellow point to be more or less the zero of another convex set. And you examine it carefully,
38:00
and you find this function you get. Turns out to, well, it's got it up there. F2s is two of f of s. It's homogeneous of degree two. Very simple. Where does the two come from? Well, the two comes from the fact that you're dealing with L2. If you took Lp, the Lp difference between one and two,
38:23
you'd get p instead of two. It doesn't matter. The important thing is you get a non-zero number bigger than one. And then, there's this homogeneity. Well, if you take two, a function like that, which is actually defined with the origin, vanishes with the origin, is actually quadratic polynomial.
38:42
It's quadratic, it's degree two. It's homogeneous, degree two. And it vanishes with the origin. Well, that function is zero. So, I've written it down for you. In other words, function is analytic, analytic is zero. In fact, it's identically zero. If f is identically zero, go back. The change of variable f to zeta
39:03
is actually a birational transformation. So, if one is zero, the other is zero. Contradiction in the proof QED. It is extremely, you see, just a few lines. All the hard work is going into this function t. That was discovered, designed for a quite different purpose.
39:22
That was designed for the purpose of physicists, to impress physicists. Now, by chance, by accident, I stumbled on this. Well, that's very nice, isn't it? You've discovered something by mistake, or not by mistake, but by doing something you weren't expected to do. Right. Well, I think I better move on fast,
39:43
since my time is running out. So, where do we go from here? Solve the great problem. Where do we go? Well, the Riemann hypothesis, called R8, can be generalized in many ways, and when you generalize it in many ways, you prove it step by step. Prove it for the first version, you'll get a more difficult version, and so on.
40:01
But there's a very subtle point, proving it step by step, it's not the same thing as proving them all at once. You have to be a magician to know this, but you think that, if you can prove everything step by step, essentially that takes you everything. That's not quite true. It depends on logic, it depends on what causes ZFC axioms, Zermelo, Frankel,
40:20
which is to do with theory, and C serves as the axiom of choice, and the axiom of choice hides many choices, and that's where things really get interesting. Then the question is, numerical computation. Riemann and Zeta, thought it was interesting, because it gave you information about primes. You actually want to know what those numbers give you. You have to compute.
40:40
That's the numerical computation, it's not the same thing as proving they exist. Now here there's a lot of work for young people. When young people are here, they hear that a problem has been solved, the thing is all done with. Not at all. It's just the beginning. Now you can use lots of people in mathematics, computer science, logic, physics, they can get to work, because lots of questions remain to be solved, and then the question is,
41:01
the most general version of Riemann hypothesis altogether, I claim, should be thought of as undecidable theorem, and that's the kind of theorems that could go to all these favors. So that's the task for the future. Beyond this, use the most powerful tools you've got available, examine every conjecture, proven or non-proven,
41:20
and decide which are effectively computable, and on what time scale. Some questions you need to know the answer tomorrow, some of you need to know it yesterday, some of you can wait for a year, some of you can wait for a billion years, if you want to know the future of the cosmos, and decide which decisions you have time to wait for. By the way, I put the word non-proven in there.
41:42
In the British courts of justice, English courts of justice, you can be found guilty or innocent. It's a binary choice. But in Scotland, which is more subtle, there's a third choice. It's called non-proven. You can be found guilty, innocent, or non-proven. That's actually much more subtle logic.
42:01
And come to a lot in Scotland if you've got a subtle case. Okay, well, I've really gone on too long. I probably re-used up my question time, but I'd better stop there. Thank you.
42:22
Thank you, Michael, for this entertaining and fascinating presentation. We have time for a couple of questions, and I guess... From the background, okay? Nobody in the front. Okay, from the background. And wait for the microphone. Don't be shy.
42:41
Go on, yeah. I'd like equality. One girl, one boy. Who wants to meet first? The girls or the boys? Competition? Come on now. Don't tell me you understood everything. Everybody's too hesitant. Yeah, there's one.
43:01
Don't be hesitant. Be brave. Wait for the microphone. This man is brave, yes. Hello, sir. So this was a fascinating talk. I come from an artificial intelligence background, not the mathematics background. So I don't understand much of the technical stuff. But yes, I am still curious after the talk.
43:21
So this talk created a lot of hype on Twitter and all that. This is going to solve an unsolved problem. So I see... So what is the concrete answer to it? Is the Riemann hypothesis solved? This is going to solve one of the Klee Mathematical Institute's seven problems, or what it is. I am still fascinated.
43:40
I am not able to absorb what is it about. Right. Well, the question was... Have I solved the problem? Can I claim the prize? Well, it depends... You see, again, it depends on your logic. The conjecture was the original conjecture of Riemann, I've solved the problem. The Riemann hypothesis has been proved.
44:02
Unless you're the kind of fancy tradition who doesn't believe in proof by contradiction. In that case, I have to go back and think again. People usually accept proof by contradiction, while there's proof of the Fermat's Last Theorem, is it proof by contradiction? So I think I would argue my case, I deserve the prize. But, but, but, but that doesn't solve all the problems.
44:22
There are many other questions that follow the generalizations of the Riemann hypothesis, which are not follow automatically. So the Riemann hypothesis conjecture is the first step on a long road. But, yes, the first step is the solution of the problem. Period. I can retire now.
44:42
But there's work for you. Now, that was a man, so we need a lady now. Don't be shy, don't let the boys beat you. My wife, who's a competitor, went to a mixed school, and she beat all the boys. Ah. Yeah. Yeah, I think you're seriously far back to count as a youngster.
45:03
I mean, the rose. And when will you put this on our side, that we can check your proof? Proof is there. Well, no more seriously. I've written a number of papers,
45:20
the longest paper is on the Feinschrätze constant. And that is written, a complete version, written, it's been submitted to a journal, a very respectable journal. But you see, when you get to be my age, people don't publish your papers, because they say you're too old, and there must be a mistake somewhere.
45:41
So it's very difficult to get people to publish my papers. But it is written, it's available, I can give you a copy of the text of the Feinschrätze. And that's the hard one. This other one is a five page query, essentially up there, and I've got a version of that too. So it's all written down. Yes, but it's hard to get it published.
46:00
I even submitted it to the archive, and they don't accept it. So, you know, that's what's called ageism. You know, you think only the sexiest are discriminated, but old people are discriminated too. And I'm definitely qualified as an old man. Okay, last question.
46:22
Two for the go. Hi sir, thank you very much. You said that nobody believes any proof of the Riemann hypothesis, because no one has ever proved it. How high are the stakes for you, that everyone believes this?
46:41
I mean, whether or not you win the prize, is it important, or are you just like, well, I did it, and I don't care who believes it? How do you feel about it personally? No, I think I do care who believes it, because as Tom said, mathematics or science in general involves two steps. First, creation, then dissemination. If you don't disseminate your ideas, they don't get anywhere.
47:02
No, I don't believe it's important, not to get the prize, but to know that somebody else is listening. But I'm short the memory gone, so what was your first follow-up question? It was...
47:21
Well, no, sorry. The other point was, I see, people will not believe a proof that you have a fundamentally new idea. If you come along saying, I've used the old methods more cleverly, that's so good, but if you come along and say, I've got a totally new idea, they should listen, and I've got a totally new idea. A totally new idea, I explain to it,
47:40
but in some sense, if you like, ultimately it's about using infinite iteration of exponentials, which is, in terms of set theory, it's a set of subsets of subsets of subsets ad infinitum, and that is a very difficult notion, but with that, with von Neumann, my backing, I wouldn't, I'd be scared stiff.
48:01
So with von Neumann on my side, I'm quite happy, and so I have got a new idea, fundamentally a new idea, so people should listen. If I just came along with old ideas, they'd say, well, it's been tried, it won't work, but this one is a new idea, so they should listen. Thank you, Michael, you gave us a lot to think about,
48:21
but maybe we should not go into the coffee break before we get the answer about Hardy's umbrella. Yes, yes, and there'll be a small cup, I'll give you a cup of coffee, you will solve the problem.
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