 Okay, so we can do a 415, so we're just going to go into five, five plus options. So welcome everyone, it's a pleasure for me to introduce Emanuel Carnero from INPA, Rio de Janeiro. So he's Simon's associate of the ECTP since last year and he will talk about dream and hypothesis, a million dollar mystery, so please. Thank you all. Thanks, Victoria. Thanks everybody from the math section for the lovely invitation to speak in your basic notion seminar. Is that okay? Can you hear me? Maybe I should. Okay. If at some point it's failing, just let me know. Maybe I can put it on my shirt. How about now? Is that better? Okay, but it's better if I speak now. Okay, so we should do good now. Now the topic of the day is one that I believe everyone is familiar with, the dream and hypothesis, right? So it's a million dollar mystery. In this talk, which will be very basic, I promise, I want to convey some of the ideas behind the problem and some little bits of stories that lie around this problem that has almost 160 years. Okay, so I should start by mentioning and by giving the proper credit to two individuals, to two very nice professors. One of them is Brian Conry from the American Institute of Mathematics and the other one is Jeff Valer from the UT Austin. I give them special thanks and I give major credit for lots of parts of this talk because I learned a lot from them and I've seen them giving talks on this subject, so some parts of this talk is heavily inspired by what I learned from these guys, which are both amazing. You can find talks by Brian and by Jeff on the dream and hypothesis and other nice topics and number theory you can find online. Okay, so everyone ready? Okay, good. I will start this talk with a quote from a famous mathematician that I like a lot. He said, and I quote, if I woke up from a 500-year sleep, the first thing I would ask is if the Riemann hypothesis had been solved. Who said that? There are the initials. David Hubert, exactly. So this is one of my favorite mathematicians. I will start my talk today with Hubert and I will finish with Hubert, but the Riemann hypothesis actually appears in the list of 23 problems, the famous list of 23 problems that Hubert proposed in the International Congress of Mathematicians in Paris in 1900. And here it is. It's problem number eight. You see here the Riemanns at the function that we'll be talking about a little bit more. As advertised, this is one of the Clay Millenium Prize problems. So in the year 2000, the Clay Institute of Mathematics in the United States launched the project to give awards, seven awards of $1 million to the solution of seven problems in mathematics. These are the seven problems. One of them is the Riemann hypothesis. Only one of them, the Poincaré conjecture, was solved up to this date. So there's six million dollars in play if you get interested. Not only in the Riemann hypothesis, but any of the problems would make you slightly richer if you want. So I start to say that my talk was going to be very basic. So my first slide, I will define the prime numbers. So the prime numbers are two, three, five, seven, and so on, the numbers that have only two divisors, one and themselves. So these are the building blocks of the integers. Number theory's purpose is to actually understand the structure of the integers and its building blocks, the prime numbers. So here we have the basic fundamental theorem of arithmetic that every integer can be written as uniquely as a product of prime numbers. Like this being a very old and a very basic statement, it is the fundamental principle behind many of the technologies that we use today for factorization of numbers with large prime factors. This is behind some sort of cryptography and the codes that we use every day when we access our bank accounts online, for example. Well, problems about primes have fascinated mankind for 2,000 years or more, right? So I'm going to quote here a few of these problems about primes that you might know. So let's say on the letter from Goldbach to Euler in 1742, and here's the part of the letter, okay, sorry, where he actually quotes in German, I say, it seems that every number larger than 2 is the sum of 3 primes. And then Euler replied on June 30 of that same year says that every even integer is the sum of 2 primes, I regard this as a completely certain theorem, although I cannot prove it. And as you know, this problem is now known as the Goldbach conjecture. So just to give you a little bit on the developments that this problem over the years, Ramare in 1995 was the one who proved that with 6 primes, you can add up 6 primes and get any number. Well, number has to be bigger than some number and zero. And Harold Helf got in 2013 proved that every odd number is the sum of 3 primes. So this is known as the ternary Goldbach problem. The original one remains an open problem and it's verified up to 10 to the 17. So every integer, every even integer is the sum of 2 primes. Here are the pictures of those guys, Euler and Goldbach. You'll see that's a nice feature of this talk that all the mathematicians that I'm going to talk about, with the possible exception of Riemann, are guys who actually live the long and prosperous life. So this is something very nice for us mathematicians that we have the chance to do this amazing job where we've actually paid to thank and discover new things. And we can do this until the very last of our days in here. All right, continuing the problems about primes. So let's talk about this twin prime problem. Do there exist infinitely many primes p such that p plus 2 is also a prime? This is also an open problem called the twin prime conjecture. The largest known twin primes as of September 2016 are these numbers. So just to give you an idea of the computational power that we have nowadays, these numbers have 388,000 digits roughly. It was just recently proved, this is 2013, that there exists infinitely many pairs of consecutive primes whose difference, whose gap is less than a fixed number. And it proved that there's infinitely many prime gaps which are less than 70 million. That's what he proved. This was refined by James Minard a few months later, who actually brought this 70 million down to 600. What I mean is that there's an infinitely many pairs of consecutive primes whose difference is less than 600. And the best form of this was achieved by this polymath project, who actually brought the number back down to 246. There's an infinitely number of consecutive primes whose difference is less than 246. The goal of this project is to bring this 246 down to 2. We want to prove that there's an infinitely many pairs of prime numbers whose difference is 2. Okay, but this is not known. So other problems about primes, there exists infinitely many primes of the form n squared plus 1. This is also an open problem. And of course the problem that has fascinated some people working in prime number theories is the problem of counting the number of primes up to certain integer x. So people have been fascinated by finding formulas that would give the counting prime function. And eventually some of us that work on these things will receive emails from some people say, hey, I found a magic polynomial that yields me the formula or yields me the prime numbers or something. Not exactly an expert on this type of CV methods and this type of problem, but I believe that there's a natural threshold on their method. Now it's not just a matter of bringing more people to work on this. There's some sort of a philosophical barrier that needs to be trespassed before achieving the two there, or even getting closer to that. Yeah, and by the way, feel free to interrupt me at any time for questions, for comments, for whatever you want, to sing, tell a joke to your friends. So let me now move and tell you some stories about some of the classical developments in prime number theory. So we start with this big guy. We start mentioning the words of Euler. So Euler was, for example, this identity that is on the board is attributed to Euler. The sum of the inverses of the squares is pi square over six. And I always found this identity fascinating. I remember when I first saw this, I must, maybe I was like 11 or 12 years old. I was fascinated how this number pi appeared there. And this is kind of, I think if you're meant to have a mathematical mind, you should ask yourself, why the hell is the pi there in this formula? It makes no sense at all. I know it's like 10 or 11 years old, but now it makes a little bit more sense. But if you don't get struck by this pi, so maybe you should look for career medicine or something, or law school. So Euler considered the function, the sum of the inverses of the numbers, right? So for x a real variable, he considered the sum of the inverses one plus one over two to the x plus one over two to the x, so on. And he could actually use the fundamental theorem of arithmetic to actually factor this sum into an infinite product. Now just over the powers of the prime numbers. So we have all the powers of two here. All the powers of three here, all the powers of five here, the powers of seven here, and so on. Of course, when you multiply all of this together by the fundamental theorem of arithmetic, you get every integer number appearing just once. So this is why this identity holds. Now, so alternatively, you can write this sum as the product of, you can actually evaluate this geometric progressions and write it like this, so you have this product formula. We use this sort of identity to show that the sum of the inverses of the prime numbers diverges, which is also already a non-trivial result. So we're talking about 1700s here. So this sort of sparsity in the counting of prime numbers, as I said, already fascinated people. So for example, so you can take a look, here's a table with the first, so this is x in the left column. And this is the quantity of numbers up to x, the quantity of prime numbers up to x. So up to 10, we have four prime numbers, up to 100, we have 25 prime numbers, and so on. So you see it, when you look at these numbers, you cannot actually guess what they're gonna look like or guess a relation among them. But when you start looking at the ratio, you mean the ratio of x divided by 4 is 2.5. Next divided by 25, this is 4, and so on. You start to get some numbers here, which start to have some pattern. And this pattern is like if you look at the difference between these ratios. This is supposed to be like an arithmetic progression. The difference of ratio seems to be converging to some number, which is gonna be a magical number. And in fact, this sort of heuristics can be made formal, right? So the number of primes up to this magic number that appears here is 2.3. And 2.3 appears there because it's log of 10 on the base e, so it's log of 10. So the number of primes up to 10 to the n should be 10 to the n divided by 2.3n. 2.3 is log 10. So it should be 10 to the n, log 10 to the n. So if you believe that this 10 to the n has nothing to do with the quantitative problem, you kind of are led to formulate this so-called prime number theorem. That the number of primes up to height x, this function pi of x that counts, the number of prime numbers up to the given real number x, is roughly x over log x. This was first conjectured by Gauss when Gauss was around the age of 16 years old, right? This was around 1792, 1793. But it took 100 years to actually be proved. You all must have heard the story of Gauss when he was like maybe seven years old, and he went to the first grade class, and the teacher arrived in one day and was a little bit bored, and just wanted to send the students to do something for one hour while she would rest. And she told the students to calculate the sum of the numbers from one to 100. And Gauss replied in ten seconds, the answer is black. And this was when he was at the age of seven, right? So this he was a little bit more mature. He was at the age of 16 to conjecture the prime number theorem in this fall. Well, as I told you, this took 100 years to be proved. There were some intermediate attempts, some in the right direction, some in slightly off directions. One that has made an attempt to this problem was Legendre in a paper in 1808, a few years after here. He actually didn't know that Gauss had conjectured this because this only became public 50 years later. Legendre claimed that this function pi of x was approximately x over log x minus a constant b, which he calculated numerically to be some number. Because he had tables of primes up to 400,000 at the time, which was already an achievement for the time. This turned out to be slightly wrong, this number b is not here. But anyway, he was very close. So this is called Friedrich Gauss, who again, as I pointed to, lived almost 80 years. He conjectured, as I pointed in the previous slide, that this function pi of x was supposed to be roughly equivalent to this integral of 1 over log t, which gives x over log x as the main term. This function is called Li of x at the age of 16, but it was only reported in the letter 50 or 60 years later. This is my own recollection of Gauss, the clothes that I have been to Gauss. People will always tell me, you're never even getting clothes to Gauss, no matter how you study. I say, well, it can be as close as it gets nowadays. This is the clothes that I have been. This is the tomb of Gauss in Gottingen, the Gottingen cemetery. So if you ever visit Gottingen, you can take a stroll in the cemetery and visit the tomb of Gauss and take a picture like me. This is a very fun thing. This is the handwritten of Gauss. This is the picture, the source here is the Gottingen Library in the American Institute of Mathematics. This is how Gauss recorded his statistics of primes. He called this a chilead. And this is an interesting mechanism that he used to count. So just that you can take a look how he did that. Here he's counting the prime numbers from 1 million to 1 million and 100,000. Yes, 1 million and 100,000. So he has a block of 100,000 things. So he divides this block of 100,000 numbers into 10 blocks of 10,000. Okay? So this corresponds to the columns, 10 blocks of 10,000. So these are the first 10,000, these are the second 10,000, these are the third 10,000, and so on. Now in each column, he divides those 10,000 in 100 blocks of 100, okay? And then he counts for every of these blocks of 100, how many primes you have on that block. So for example, and these are the numbers in this column here. So in the first block of 10,000, there was one block of 100 with one prime. There were two blocks of 100 with four primes. There were 11 blocks of 100 with five primes, and so on. So the sums of all the columns here should be 100 because you have 100 blocks of 100. And when you multiply each of this number here by the number of primes, you get the total count of primes in that interval. So for example in this, yes, he was getting from hundreds and hundreds and everything by hand. Yes, and then getting in these tables. So for example, you see that in this interval, he gets the number of primes 7,210. And here is a little sketch of his integral, the integral of one of our log, which is supposed to be 7,212. So it's kind of very close in this interval. This is very curious, right? Here is a manuscript of a letter from Gauss to his former student Enke, where he actually mentioned this. So here is the first lines of a four page letter from Gauss to his student Johann Enke, Lieutenant of artillery, dated December 1849, the letter. My distinguished friends, your remark concerning the fragrance of primes were very interesting to me in more ways than one. You have reminded me of my own endeavors in this field, which began in the very distant past, 1792, 1793, 60 years later. Enke had written to him saying that he had found something about prime numbers, and he essentially replies, well, yes, I have found the same thing 60 years before. And this is the formula for the primes. I think in the first moment, it was empirical. Yeah, they have statistics, they have the number of primes from up to a million, two million, and then more. They would start counting and that appears as a natural conjecture, which then, well, started to have some sort of support behind it. But at the first moment, I thought this was, I mean, a statistical low guess. But this is just my own guess. I don't have any actual evidence to do that. You have to go back in time and ask these guys. Chebyshev, another guy who gave a further step in this theory, gave this very nice idea, which is still used nowadays with very much effectiveness. Instead of actually counting the prime number function, this pi of x, which actually adds one every time you pass over a prime. Chebyshev had this nice idea to consider some sort of modified or weighted function. So he defined this function here, psi of x, which every time you pass through a prime power, you add log p. So every time you pass through a power of the prime p, you add log p to the sum. So believe me, this is an easier sum to handle in the sense that it's smoother. And he actually showed this equivalence that the asymptotic for pi of x being Li of x is equivalent to the asymptotic of this function psi being exactly x. And with some identities, and this is how this sort of sum becomes useful because you have some sort of numerical number of theoretical identities. He was able to prove at that point in time that his function psi lied. Remember, it has to be asymptotically equal to x. It lied between 0.9 and 1.1. And therefore, it transferred to the function pi as being between two constants of the life function. So in some sense, at this point, the function pi of x was proved to be of the order of magnitude of Li of x. But not exactly with constant one, which was expected. This is Chebyshev. And I was always curious by the number of ways that the name Chebyshev is written in the literature. So you'll see if you look closer on the Wikipedia or on other books and so you'll see perhaps like 10 different ways of writing Chebyshev. Some start with a t, some start with a c, and so on. But these are all the same guy. And I once asked a friend of mine who's Bulgarian, who understands the Cyrillic alphabet. And he says that, yeah, his original name is supposed to be in the Cyrillic alphabet, but it gets translated to different things, like slightly different. So this is why we have a myriad of ways of spelling Chebyshev. Now we move to our main guy, Bernard Riemann. He was the one who lived the shorter life of all these mathematicians that I'm talking about today. He lived 39 years. He was born in Germany. He actually died here in Italy. His grave is not exactly far from here. It's somewhere north of Milano. I don't exactly remember the name of the city here. This is a picture of his apartment that is in Göttingen. So in Göttingen, close to one of the central squares there, there is a little apartment where Riemann lived, and they had this nice plaque there indicating that's where he lived from 54 to 57. So he was a student of Gauss in Göttingen. As you know Riemann is famous for his many contributions in maybe mainly remaining geometry in the Riemann integral. And he has one paper, just this one paper in number theory, which perhaps turned out to be one of the most influential works of all time in number theory, perhaps the most influential work. It's an eight page paper on number theory entitled Jürgen Anzal, der Primzale in der gegebenen Große, forgiving my German, which essentially means on the number of primes below a given number, something like this. This was published in November of 1859. So we're about to complete 160 years in which he considered the function, the Riemanns at the function, is the sum of the E versus 1 over N to the S, where S is a complex variable. So he's inside, Euler had defined this function for S being a real variable. So his insight was mainly to consider this X a complex variable and start to make use of the complex machinery that was available at the time. So he knew that there was this product formula that was convergent in the right of one. And in this paper he briefly sketches his new discoveries about these complex value functions that he calls zeta of S. And the first nice discovery is that he finds that this function z of S, initially defined only when real part of S is bigger than 1, has an analytic continuation to the complex plane with a simple pole at 1. So this function blows up when you take S equals to 1, because we know that the sum of the inverses of the numbers diverge. And this is the simple pole of this function. This is, he found also a functional equation for this function z of S. And the functional equation takes this nice form. So if I define this function c of S being 1 half S, S minus 1 pi to the minus S over 2 times the gamma function times z of S. This function c of S obeys this very easy rule, c of S is c of 1 minus S. This allows you to go from S to 1 minus S very smoothly. He had a formula for the number of zeros n of t, the number of zeros of the zeta function up to a height t. So this is the number of zeros that have ordinate from zero to t. And he already knew that n of t was asymptotically equal to t over 2 pi, log t over 2 pi minus t over 2 pi plus a term which is of lower order. And he had a nice way to express this function, psi of x, that was considered by Chebyshev before, as a certain sum over the zeros of his functions. So psi of x was x, and this is the main order term. This is what it should be, minus something which is a sum over the zero. So he has to work on this to prove that this is of lower order than that. This was just formally proved by von Mängel in 1895. But Riemann had this formula stated in his paper. And the final statement of his paper, which is actually his hypothesis, we can see that this Riemann's function has no zeros on the real part of S bigger than 1, essentially because it can be written as a product of things that is non-zero, so product of things that are non-zero is essentially non-zero. On the left of zero, I mean on the real part of S less than zero, it has this so-called trivial zeros at the even integers. These trivial zeros, they come to cancel with the poles of the gamma function in the functional equation. So that's why we have the zeros of the zeta function at those points. And he conjectured that all the complex zeros of zeta that now we have restricted to the strip real part between zero and one, they actually have to have real part equals to one half. This is the Riemann hypothesis. This is a version of his manuscript where he submitted to the journal in 1859. So this is the handwriting of Riemann. Here you will see the definition of the Riemann's zeta function here. And here you will find a piece on the paper where he actually suggests the Riemann hypothesis. This is the German version. This is a rough translation to English. One now finds indeed approximately this number of roots within these limits. And he's referring to his zero counting function there. And it's very probable that all roots are real. When he says that it's very probable that all roots are real, he's already working with some sort of rotation, rotation version of the function to make this line, this critical line into the real axis. And he's saying that it's possible that all the roots are actually aligned in this sense. And so this simple phrase, I mean, generates a lot of research in number theory, physics that we have nowadays. This is the published version of the paper, published in November of 1859 in the monthly notices of the Berliner Academy of Sciences. You will see that the editor was actually Herr Enka, which we've seen the letter of communicating to Orla before, to Gauss before, I'm sorry. And this is a really nice paper. If you have the chance to read it, as I said, it's an eight page paper. So it's nice to take a look when you have a chance. Well, as you know, the Riemann hypothesis started on that date. It has been an open problem for the last almost 160 years. A lot of interest come also from the equivalent formulations or the consequences of its validity if established. But it's nice to see that it has some very nice and sometimes easy arithmetic equivalences that might be a little bit misleading, right? So for example, each of the statements below is actually equivalent to establishing the Riemann hypothesis. For example, if you can work with the prime counting function, pi of x, and you can prove that it's equal to the integral l of x plus an error term, which is essentially square root of x log x, then you win. Or if you can work with the psi function, which is just summing log p every time you pass through a prime power, if you can prove that it's x, that's the main term, and that the error term is of the order square root of x log square x, then you win as well. Or in this very nice formulation here, if you define the sum of the Mabius function, so this Mabius function is a function which is very simple. Mabius function of a positive integer n is going to be zero if the number n has a square in its factorization. And if the number is square three, if it has every power in this factorization raised to the power of one, then it's gonna be one if the number of prime factors is even, and it's gonna be minus one if the number of prime factors is odd. So it's reasonable to believe that in nature, there are not more numbers with a prime quantity of prime factors, an even quantity of prime factors, than a number with an odd quantity of prime factors. This should balance somehow. And this is actually stated in this formulation. So if you sum the Mabius function of n up to a height x, there should be cancellation between this one and minus one, and you win if you can get this square root cancellation here. So this is also equivalent to the Riemann hypothesis. And it's very interesting to say that some of this, one person that you might have heard that gave serious attempts at proving the Riemann hypothesis was the British mathematician Hardy. And he was famous for assigning this sort of the Riemann, assigning it to his PhDs to these graduate students in these disguised forms. Not because he was a mean guy, but because he actually believed that there's some kind of ingenious ideals necessary. So he would give this formulation to his students, say, why don't you take this Mabius function, which is one and minus one, and try to add them up and try to prove that it's as big as square root of x. And the graduate student would be very happy with such a nice problem that his advisor gave to him. He would go back home and think about it and come six months later and say, I haven't made no progress. And he would say, yes, I would expect that, but I wanted to give a try. This is equivalent to the Riemann hypothesis. But I don't know, when I think about that, this is a better, maybe this is a better way than giving your student and saying, hey, there is this problem, the Riemann hypothesis, it's been open for a hundred years, Gauss tried, Riemann tried, Chebyshev tried, and everybody else tried. Nobody did it, but you're gonna do it. So why don't you take a look? So there has been a lot of numerical implementation to support, to give evidence to support the Riemann hypothesis. So nowadays we can actually show that the first, what number is this, a million, billion, the first 10 trillion zeros of the Riemann's other function lie where they should lie, lie in the critical line. They have a real part of x equal to one half, right? This computer advances. So you see like in 1953, the Turing was already able to prove that the first thousand zeros were in the critical line. How do you prove such a thing? How do you prove such a statement? Well, this is a complex analysis problem. You can use this sort of argument principle to actually compute the number of zeros in a certain region with some precision. You say that there is a thousand zeros in this region. And then on that region, you start walking around the critical line and using some sort of mean value theorem to find positive value and a negative value, and then it has to cross once. Then it has to cross again and cross again. And you actually find those 1000 crossings that were supposed to be there. They are all in that critical line. So these are like the 1000 zeros that are up to that height. People have been refining these things, and this is the limit of the computer power that we have nowadays. Well, if you show any of these things to a people who doesn't work in mathematics, the guy would probably say, well, isn't it proved that you proved that the first 10 trillion zeros are in the line? What else do you want to prove? But for us, mathematicians like proving that the first 10 trillion in our line doesn't necessarily mean that there will be one of them out or not. Some interesting facts about RH. So as I said, the mere fact that z at the line one, z of one plus IT is different from zero. So remember, we want to prove that all the zeros are on the line one half. But the mere fact that the line of real particle to one doesn't have any zeros. This was proved by Hadamard and the Lavallet Poisson independently in 1896, and this is equivalent to the prime number theorem. So this is how they proved the prime number theorem that pi of x asymptotically is equal to x over log x. In 1948, Zellberg and Erdrich gave an elementary proof of the prime number theorem, elementary in the sense that it did not use any sort of complex analysis. It just lived in the real world. Hardly a little would use to assign the ribbon hypothesis to the graduate students. I told you that in this guy's form. Hardly was the first one to show that there are infinitely many zeros on the critical line. It was one of the hardest results. Infinitely many zeros of the set of function on the critical line. Up to this date, it was already checked that the first 10 trillion zeros are on the critical line. Zellberg in 1942 proved that a positive proportion on the zeros line on the critical line. So he had a method. He knew that up to high T, there is a certain number of zeros and he was able to prove this qualitative result that, well, when T is large enough, the number of zeros that lie on the critical line is a positive proportion of the number of zeros that are up to that high T. He had a student mean to actually make his computations effective, to actually find his proportion and the proportion that he found was one over 15 million. And he jokes that if the name of the guy was mean, he found such a small number. His name were max, maybe he would get to get to a bigger number, but it's just a folkloric joke. Brian Conrie, which is the guy that I showed in the beginning of the lecture in 1989, has the record for this proportion. He proved that at least 40% of the zeros lie on the critical line. This has been slightly improved by himself and co-authors. I think now that the record number is 41%. This guy jokes that he should be entitled to 40% of the prize, as well, but it doesn't work like that, right? So the Riemann hypothesis, as I told you in the beginning, is worth $1 million since the year 2000. Question so far. Are we okay? All right, so here's a picture of Zellberg. Here's a picture of Brian Conrie. This is actually playing cards at our house when I was a postdoc at the Institute for Advanced Study there in Princeton. This is very funny, the story of how I met this guy. So they had a thematic program in analytic number theory that year. So a lot of big shots were there, right? So we're talking Brian, Bombieri, Bourguin, Sander Aveshan, and all the greatest names in number theory that we have in the current mathematics. And I was a mere postdoc, right? So we had no chance or nothing to talk to, to most of these guys. And there in the Institute, they organize a lot of side activities for the families of the people who are there. So there are like some sort of several sports lessons, tennis and basketball and volleyball, especially for people who are bringing their spouses for them to have some activities to get engaged in the community. And there was one of these classes which was ballroom dancing. And my wife was always calling me, hey, let's go to the ballroom dance class. And I said, yeah, no, today I can't and so on. And then next week, it was once a week, let's go to the ballroom dancing classes. Well, maybe not this time, I'll be tired and so on. It went like that for the whole semester. And maybe by the end of this semester it was like the last class of the semester, she convinced me I had no, I had run out of excuses and okay, so let's go. And there in the ballroom dancing class, there was one other mathematician. It was this guy. And he kind of recognized me because we were walking in the corridors every day together and said, hey, your wife made you come here as well. So this is how I met him. So a little bit of jokes as I told you. Hardy is known for being obsessed with this problem for a little while. And this is a part of a book which is attributed to Hardy where he lists his new year's resolutions. So you can take a look at his new year's resolution for that year and see what he was accomplishing to do. So his first resolution is to prove the Riemann hypothesis. His second resolution here is something like crazy in sports, maybe or cricket, right? You should explain to people what this sort of achievement is comparable to. The three, find an argument for the non-existence of God which shall convince the general public. Four, be the first man at the top of the Everest. Five, be proclaimed the first president of the Soviet Union, the Great Britain and Germany at the same time. And number six, he wanted to murder Mussolini at the time. Maybe I should not tell this in Italy but this is not me, this is Hardy. So, Polia, Eigen values of a self-adjoint operator. So it's a folkloric way of thinking about the zeros of the Riemann's other function, of folkloric hypothesis, is that they should be the zeros of some sort of self-adjoint operator. So one potential approach to prove that the zeros are all aligned in this formulation when you do a rotation of the axis and make the critical line into the real axis is to actually find some sort of self-adjoint operator that appears naturally connected to this problem such that the zeros are their Eigen values and by the spectral theorem, if you have a self-adjoint operator, the Eigen values should be aligned, should be all real by the spectral theory. So the sort of philosophy is attributed to Hubert and Polia who were essentially, maybe at some point in their lives, tried to use this approach. This is a little letter from mathematician Andrew Woodlisco which is famous for some of these numerical simulations. Writing to Professor Polia about this matter, dear Professor Polia, I have heard on several occasions that you and Hubert had independently conjectured that the zeros of the Riemann's other function correspond to the Eigen values of a self-adjoint Hermitian operator. Could you provide me with any references? Could you also tell me when this conjecture was made and what was your reasoning behind the conjecture all time, at that time? And here was what I find interesting. This is actually the answer from Polia to Woodlisco. So Polia, he lived from 1887 to 1985, so he lived almost 100 years. He lived 98 years. So this was written in January 3 of 1982. So he was 95 years when he wrote this letter. Soon, with 95 years and still doing mathematics, that feels pretty good, doesn't it? So dear Mr. Woodlisco, many thanks for your letter of December 8. I can only tell you what happened to me. I spent two years in Gottingen around the beginning of 1914. I tried to learn as much as analytic number theory as I could from Landau. He asked me one day, you know any physics, don't you? Do you have any physical reason to believe that the Riemann hypothesis should be true? This would be the case, I answered, if the non-trivial zeros of the XC function were so that connected to a physical problem that the Riemann hypothesis would be equivalent to the fact that all the eigenvalues of a physical problem are real. I never published this remark, but somehow it became known and it's still remembered. Best regards, your Sicilian League, George Polia. Pretty nice, isn't it? You know, let's move on now to some more modern takes on the Riemann hypothesis. Well, this is what I'm gonna talk about now. This essentially, it was never proved anything on a point-wise level that the Riemann'seta function is associated to a certain magical operator such that it zero corresponds to the eigenvalues. So this would be the dream of these guys, right? And then people try to look for this statement on average. So on average, if you take some sort of average of the zeros of the Riemann'seta function and some sort of average of the zeros of the self-adjoint operators, in some sense it turns out that these two averages are the same. So this statement is essentially true on average and by on average is this connection with the theory of random matrices that I'm gonna talk a little bit now. So this is the connection between this topic in number theory and some physics, some quantum physics. So this is the zero counting function. I'm just gonna add one. Every time I get the zero with a coordinate between zero and t, so this is t log t over two pi. This is a well-known asymptotic for the number of zeros up to height t of the Riemann'seta function. Now what's in blue here is what we call the pair correlation function. So if you have this certain number of zeros up to height t, it makes sense to say divide this by this factor and you count what's the average spacing between zeros, right? Okay, so if you are from height zero to t and you want to count what's the average spacing between two consecutive zeros. So you should divide by the number of zeros on that interval and you should have the average spacing if they were equally spaced. But as it turns out, they are not equally spaced and this is what this pair correlation function measures. So here I'm counting the numbers of pairs of zeros up to height t whose difference is less than beta times the average spacing, okay? So whose difference is less than beta times the average spacing. Okay, so this is what it's. In 1972, Hume-Mongolmery conjectured that this pair correlation function should be that the zero counting function times a certain integral of a certain kernel, this kernel one minus sine pi x over pi x squared. Now let me point out that this is exactly the measure of irregularity that the zeros should have. If this sine pi x over pi x squared were not here, if this was just an integral of one, and then this would mean that the zeros would be equally spaced on average. But this is not, there's some sort of skewed scheme here such that the average spacing of zeros corresponds to this sort of density function. And this is still a conjecture called pair correlation conjecture. Now, there's a nice story about this, how these two fields came together, number theory and random matrix theory. And this happened in the meeting in the spring of 1972. This is the Institute for Advanced Study in Princeton where these four guys played a role in this story, right? Here we have Mongolmery, here we have Chala and Zellberg were mathematicians working at the Institute and we have Freeman Dyson, the famous physicist that worked there at the time. So, at that time Mongolmery was a postdoc, he was maybe 27, 28 years old, he had just come back from Cambridge to the United States and he had a conversation with Dyson to show a little bit of his work and Dyson made the realization that the same sort of density function that he had, that function one minus sine pi x over pi x squared was appearing in some works, some eigenvalues of random Hermitian matrices in this Gaussian Unitary Ensemble Model. So, at that point, this is what made this connection possible between these two fields and let me tell you a little bit of this story as described by journalist Sabat in this magazine, Atlantic 2002. So, let's take a look together. So, this is human Mongolmery talking. Daman Chala said, have you met Dyson? And I said, no. And he said, I'll take you to introduce you to Dyson and I said, no, no, that's okay. I don't need to meet Dyson. This went back and forth until it ended up with Chala dragging me across the room. I didn't really want to bother Dyson. I didn't think of having anything useful to say to him but when Chala introduced me, Dyson was very cordial and asked me what I had been working on. So, I told him that I had been looking at the zeros of this other function. It was when Mongolmery mentioned the formula that he had found for his distribution that Dyson's ears sprit up. At the mention of the function 1 minus sine pi u pi u square, Dyson said something like, well, that's the density of the percorrelation of eigenvalues of random matrices in the Gaussian Unitary Ensemble. This is continued. And then Mongolmery says, I had never heard of these terms before. I don't know exactly what his words were because I had heard all of these terms many times ever since but he said percorrelation and he said something about random matrices. So, what Dyson had spotted was a connection between two apparently unconnected fields of knowledge, quantum physics and number theory. It turned out the physicists looking for ways to characterize the behavior of atomic particles had come up with a formula that was very similar to Mongolmery's description of the zeros of the Riemann set of function. I asked Mongolmery what he had done much of that, up to that point, but that mathematical entities mentioned by Dyson, known as random matrices. Mongolmery replied, I had never seen a random matrix, he said, and I have hardly seen one ever since. Furthermore, for Freeman Dyson, this t-time exchange seems to have been just momentary diversion from his own very different line of study. As far as I know, Mongolmery says, he didn't think about it after this five minute conversation. I haven't spoken with him since, so I've had one conversation with him in my life for five minutes, but it was quite fruitful conversation. It happened just at the right moment because I had this result and what was needed was the connection and he provided the connection, but it didn't alter the mathematics, it altered our understanding of what the mathematics was related to. I suppose that by now somebody else would have found the connection. It's nearly 30 years ago, but it certainly was from the standpoint of publication, instantaneous, I had the mathematics and as soon as I had it, it was just a matter of months before the connection was pointed out. From that conversation has come a whole new approach to the Riemann hypothesis and the possibility that in some quite significant way, the quantum universe behaves as if driven by the location of the Riemann zeroes. It's a very nice story. So here is actually the letter after they had this conversation. So the letter is dated April 7th, 1972 and it's a letter from Freeman Dyson to Atle Zellberg who was the mathematician-in-chief of the number theory section at the institute. Dear Atle, the reference which Dr. Mongolmery wants is, blah, blah, blah, blah, blah, showing that the percorrelation function of the zeroes of this. That the function is identical with that of the eigenvalues of a random complex Hermitian-oriented matrix of large order, Freeman Dyson. You'll see the amateurship of the guy taking the picture, right? Because it's me. So this letter is in human Mongolmer's office. He has it framed and when I visited him, I asked to take a picture, so that's why it's so amateur. This is not my professional at all. But it's very nice. We see the date. So the conversation took place on this date which is significant for mathematics and physics. And this is my own recollection with Mongolmery in 2015, in Michigan. Anyway, I told you that I was gonna start this lecture with Hubert and I was gonna finish with Hubert. So let me just conclude here by saying a few words on this. David Hubert was also an enthusiast of the Riemann hypothesis, as shown by his quote in the first slide of this lecture. There is a famous story. I don't know if maybe someone heard this before, but if you heard it, it's always nice to hear it again. And if you haven't heard this before, it will be, I hope it will be as shocking to you as it was for me when I first heard this. So on September 8th of 1930, David Hubert addressed the yearly meeting of the Society of German Natural Scientists and Physicians. He was retiring. He was at, with the 68 years of age. He was retiring from his professorship at Goettingen and he was being honored in his home city of Konigsberg. Hubert, in his speech, he delineated his basic research philosophy, that every mathematical problem is solvable. And he encountered at that time a widespread opposing opinion. There were people in that particular moment in the world, in that particular moment of science who were just happy enough and content enough to admit that they were ignorant and there were things that were bigger than us and we should not pursue these things. So they were happy with their own ignorance which Hubert himself called the ignorabimus. So some people were happy and were okay to live with acknowledging the ignorabimus, but he was not. Shortly afterward, he read on a German radio a four minute version of the finale of his speech. And this is what I want to play to you now. So if you have never heard, this is the first time that you will hear David Hubert's voice on this very nice speech. Let me see. And I want to see if I can make this big. No. How can I make this big? Das Instrument welches die Vermittlung bewirkt zwischen Theorie und Praxis, zwischen Denken und Beobachten ist die Magnatik. Sie baut die verbindende Brücke und gestaltet sie immer krachfähige. Daher kommt es, dass unsere ganze gegenwärtsche Kultur, soweit sie auf der geistigen Durchstrimmung und Dienstbarmachung der Natur beruht, ihre Grundlagen in der Mathematik findet. Schungali Lee sagt, die Natur kann nur der Verstehen der ihre Sprache und die Zeichen kennengelernt hat, in der sie zu uns redet. Diese Sprache aber ist die Mathematik und ihre Zeichen sind die mathematischen Figuren. Kant tat den Ausspruch. Ich behaupte, dass in jeder besonderen Naturwissenschaft nur so viel eigentliche Wissenschaft angetroffen werden kann, als darin Mathematik enthalten ist. In der Tat, wir beherrschen nicht eher eine naturwissenschaftliche Theorie, als bis wir ihren mathematischen Kern herausgeschehen und völlig enthüllt haben. Ohne Mathematik ist die heutige Astronomie und Physik unmöglich. Diese Wissenschaften lösen sich in ihren theoretischen Teilen geradezu in Mathematik auf. Diese, wie die zahlreichen weiteren Anwendungen sind es, denen die Mathematik ihr ansehend verdankt, soweit sie solches im weiteren Publikum genießt. Trotzdem haben es alle Mathematiker abgelehnt. Die Anwendungen als Wertmesser für die Mathematik gelten zu lassen. Gauss spricht von dem zauberischen Reiz, den die Zahlentheorie zur Lieblingswissenschaft der ersten Mathematiker gemacht haben. Ihres unerschöpflichen Reichtums nicht zu gedenken, woran sie alle anderen Teile der Mathematik soweit überdrückt. Chroniker vergleicht die Zahlentheoretiker mit den Lotophagen, die, wenn sie einmal von dieser Kost etwas süßigen Namen haben, nie mehr davon lassen können. Der große Mathematiker Poincaré windet sich einmal in auffallende Schärfe gegen Tolstoy, der erklärt hatte, dass die Forderung die Wissenschaft der Wissenschaft wegen töricht sei. Die Errungenschaften der Industrie zum Beispiel hätten nie das Licht der Welt erblickt, wenn die Praktiker allein existiert hätten und wenn diese Errungenschaften nicht von uninteressierten Toren gefördert worden wären. Die Ehre des menschlichen Geistes, so sagt der berühmte Königsberger Mathematiker Jacobi, ist der einzige Zweck aller Wissenschaft. Wir dürfen nicht denen glauben, die heute mit philosophischer Mine und überlegenen Tone den Kulturuntergang prophezieren und sich in dem Ignorabimus gefallen. Für uns gibt es kein Ignorabimus und meiner Meinung nach oft für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung Wir müssen wissen, wir werden wissen. These final words, wir müssen wissen, wir werden wissen, are the words written on Hubert's grave, which is also in the Göttingen cemetery. And this is what I would like to be my message to you in the end of this lecture, that we pursue his goal in whatever science we are doing, mathematics, physics, astronomy. We do not rest until we have the answers we seek. We must know and this means that we will know at some point. Thank you guys so much for the attention.