4th HLF – Laureate Lectures: Ngô Bảo Châu
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Châu, Ngô Bảo
Ngô Bảo Châu: “The Riemann zeta function and its generalizations” Since the publication of Riemann’s memoir on prime numbers less than a given magnitude, the zeta function has never ceased to fascinate mathematicians. We will discuss its many appearances and generalizations in number theory. The opinions expressed in this video do not necessarily reflect the views of the Heidelberg Laureate Forum Foundation or any other person or associated institution involved in the making and distribution of the video.
 So please come in, have a seat, so that we can start our second part of the morning. My name is Anna Wienhard, I'm a professor here at Heidelberg University and a member of the scientific committee. And it's a great pleasure to introduce our next speaker, Nobel Chow from the University of Chicago. As, I mean, all our laureates, of course he won the FIATS medal as he has won several other awards. I think he has something which is very untypical even among the FIATS medalists, so his proof of the fundamental lemma made it into the Time magazine's 10 discoveries of the year in 2009, the year before he was awarded the FIATS medal. And I remember being at lunch at the Institute for Advanced Study when Peter Sarnak, a permanent member in the School of Mathematics, came to the lunch table, very excited because he had just received a call from the Time magazine asking him what he would think of being very important discoveries in mathematics. And he said, well he told them about the fundamental lemma and he really hopes that it will make it into the Time magazine. And so it was number seven of the 10 discoveries of the year 2009. And I think the fundamental lemma of course fascinated people in mathematics but apparently also outside of mathematics. Today, from Bautro, we'll give us another introduction and talk about another fascinating topic, the remensate of function and its generalization. Thank you. Thank you very much for the introduction. It's a pleasure to have you back here and to give this talk on remensate functions. I'm not going to talk about fundamental lemma, although at some point I'll say what way it can be used for at some point in my talk. So my talk about this remand data functions, the remand wrote one single paper in number theory, but that is probably the paper who had the deepest influence in the whole development of this theory. And I try to convey a sense why number theories are so educated by this quite fascinating object that is remand data functions. But let's step back a little bit into history. This question that has been addressed as early as the Greek times in Orchid. Orchid made this extraordinary proof that there are many prime numbers that I guess one of you know by contradictions. I'm not going to repeat that proof. Rather I try to explain in a non-regulatory way another proof that due to Euler that he's somewhat less known. So Euler considered this series, you know, sum of n to the minus s as a function of the real variable s. Of course this series only converges for real numbers as greater than one. It diverges as to how many series diverge. But what's really interesting is it had this factorization. On the left hand side is a series on the right hand side is infinite product over the prime numbers. You have this expression because of the unique factorization of integers. Every integer can be factorized in a unique way as for the primes. And that fundamental property of arithmetic can be encapsulated in this single identity. So it can be an analytic expression of that fundamental theorem arithmetic. So of course this bold side makes sense only for s greater than one. But now look at what happened at s equal to one. On the left hand side you get the divergent series. And the right hand side it will be convergent if the set of prime numbers is finite. Every single factor has no point that s equal to one. So it has only finite product and there will be a problem. So that was the observation of Euler. So that is Euler proof. So that is some of this long series. So this is the data function. It's really data functions. But Riemann instead of considering the function of real variables, Riemann considers this as a form of complex variables. And this stuff brings in a huge chain of landscape. So now you have this series, this functions. It converges if the real impact of s is greater than one. And of course this can be proven easily that it is absolutely convergent uniformly on compact time and so on. So this series, this whole series converges to very nice functions, holomorphic functions on the domain of real impact of s greater than one. And moreover the Euler arguments in the plies. That means these data functions on this domain have this factorization in product of factors indexed by prime numbers. So this argument of Euler extends to the complex domain. And so what Riemann is, he proved that this function now can be extended on the whole domain of complex plane. But it catch poles at s1 to 1. So it's the only single pole at s1 to 1. And in that way he can make sense of these kind of funny identities when you put s1 to 0. So of course this doesn't make sense. This is 1 plus 1 plus 1. But then if you compute the value of this function, it's not completely fine. And you get zeta of 0, you can 1 minus 1 half. And sometimes 1 plus 1 plus 1 is minus 1 half in Riemann's sense. And if you put s1 to minus 1, it's even more funny. So 1 plus 2 plus 3 would be minus 1, 12. So that is... Anyway. So these papers have had a huge influence in number theory because it set a way into understanding prime number theorem. So how many are their primes less than given magnitude? So the argument goes as follows. Here you have zeta. Zeta is a product over primes. If you take a logarithmic derivative, right? And then product becomes sum. So you get... This is a logarithmic derivative of zeta. And you can develop into a sum. And what happens is it's almost a sum over prime numbers. So this lambda n is called von Machen functions. Lambda n is a log d if n is a power of prime numbers. So if lambda of 60 is 0, for example. So basically when you take a logarithmic derivative, write... You have written it as series. You only get... Only left the coefficient corresponding to prime numbers. This is actually sum over primes. So if you didn't know pi x, the number of primes less than given real number x, then the prime number theorem that was actually already suspected. I mean, hypothesized by Gauss and Legendre by numeric calculations. Pi x should be equivalent to x over log x. x has to go to infinity. And pi x is basically a sum of 1 plus 1 over the set of prime numbers. Now, because there's not so much difference between the prime and the set of power primes, it's not... It's quite elementary to prove that this... The prime number theorem is equivalent to this. So instead of counting prime, we just sum the von Machen function up to n less than given x. And if it's equal to say this sum is equivalent to x, then x goes to infinity. So this requires an argument, but not... And we talk to you with those arguments, but it's very elementary. So now we see the relation between the prime number theorem with von Machen functions and with Lorentz derivative data functions. So this is... When you look at Lorentz derivative as the series, you get this lambda n as coefficient of the direct-clase series. And now the question is how to estimate this sum of lambda n. Okay. So now we come a little bit of harmonic analysis. So let's look at some questions on the direct-clase series. Let me just sum n to the s with coefficient of some given function. You can be von Machen function can be anything. And with some... Now, you really see why it helps to look at s as complex parameters, because it can use complex integrations. Instead of having only real and lie, you can integrate your vertical lines. And so now if you get this function, you integrate it over some vertical line. So let's assume that this series is converted absolutely, right? When s... When a real and particular s greater than sigma zero, some numbers. So we have nice convergent series on some half planes. And on that half plane, you can estimate this sum, right? a n, this part sum sum, a n, n less than x by these co-integrals of vertical lines. So it looks a little bit complicated, you know, but the very basic things, you know, if you don't... if you only care about x. So the only occurrence of x is actually the s, right? So if you ignore all the difficulties, we'll say that this is basically actually the s, because that is the only thing that counts, right? So you have estimate, you can estimate the sum of the left by some constant time, actually the sigma. So that's, you know, very roughly the basic information you can get from this complex integration. But now this should not depend on sigma, right? Because it... So you get better estimate by moving your line integration. As long as the line integration is contained in the convergent domain, there's no difference doing that. And you can move in integration on the way where you get some points, right? So I mean, you get some points, I mean, in comparison, it's not a problem, just move across a point and you catch the residues, right? And just do... doing so by doing... by this trick of... this is very basic tricking as number theory, just moving the line integration further to the left as much as you can, right? And that you can get some kind of asymptotic expression of this sum, right? The main term should be the residues that you catch along the way moving your line integrations, right? And you can do it as much as you can analytically, and continue your complex function. So that's why it's very important to get the meromorphic continuation of the data function, right? And when you get it, you can also meromorphic continuation of this locality derivative and so on. Okay, so now we want to understand this... to apply this trick, this complex integration to this... the locality derivative of data, right? And so we need to understand, you know, anyone... then you say the lambda 1 plus lambda 2 plus etcetera plus lambda n can be estimated by the points of the locality derivative of data. But the point of this is exactly the points and the zeros of data. So that is why you really need... data doesn't have... to have one point, it's very simple. To put it one. So what do you call as zero data function? So that's why it's such a big deal to understand zero data functions. So for example, by following this... by the strategist Adam M. M. de la Valle Poussin, prove rigorously at the beginning of the 20th century that there is some... you can extend slightly over the line... So I have a real property of s1 to 1, so I have some zero free zone. You know, you get closer and closer to 1 when the imaginary part gets to infinity, but it's really the open part. And just with this, they can prove the pi number theorem. Yes, lambda 1 plus lambda 2 plus lambda n is equivalent to n. But of course the demand condition is much stronger. He's... the zeros are on the critical line of real property of s1 to 1,5. And somehow the estimate can never go past that line. That is the best you can do, which is go closer, closer to real property of s1 to 1,5. Okay. So that I tried to say in very rigorous way ideas in demand papers and the development further. Of course there's a lot of difficulties, I ignore all the possible energy difficulties in doing this, you know, inverse Fourier transform and so on. But now let's... one of the... I think it's very remarkable if this storage doesn't apply only to Riemann data function, but to many similar functions. So one of the basic examples that come on mode at the same time as Riemann is this Conductor L function. So it's very similar. But we have some more data. So you start with some n, Conductors is natural numbers. And you are giving a characters of the multiplicative group Z and Z to C star. So let's pick an example at n or 2,3. And so all we have one, two class that is... so Z mod 3 has three classes, modulo 3. And the cross just delete this zero. So the characters can pick one when n is going to 1 modulo 3 and minus one when n is going to 2 mod 3. And so form this function on the class from the modulo n, you can just extend it to Z, just put in zero where the function is not defined earlier. And then you can just form this series. So it's completely similar to the Riemann data functions, but instead of one, you put one or minus one, you know, or unit roots, depending on your characters. And of course, if n, if here, big n, 1, 2, 3, then there's no more term with n, 1, 2, 3. So term is just left out. And very similar to Riemann data function, you can improve this function. It has metamorphic continuation. And actually it's more, quite amazing. It has no bones. If the guy is not trivon characters, there's no bones. And that is how directly prove that there's infinite primes in arithmetic progressions. So that is, you know, you can count the number of prime, you know, arithmetic progressions by the same kind of tricks, you know, and just apply the Perron, this is called the Perron formula, to this directly L functions. And you get the into that. There's no, and moreover, they don't have no bones. And also, it is, this L function is supposed to satisfy Riemann hypothesis also on this interesting zeroes, must be on the lie of Riemann data, one half. But of course, nobody have a single clue how it's true. Okay. Now, let me move on further. So as in, as was emphasized in S. and Riemann lectures, number three had to move at some point from a billion to non-nibbillion. And here the, the break point is the theory of, or HECA theory of L form attached to modular forms. So, I mean, I haven't studied enough of history to know, but I guess this, that I was not clear why it is connected, even connected to Riemann data function, although it has many similar properties. So, history is called modular forms. Modular form is essentially some kind of whole-on-mode function on the, on the half plane, right? And this satisfies a modular relation. So, like, AZ plus B over CZ plus D is, it doesn't change the function up to some, some factors. So, this K is the weight, the weight of the modular forms. And this function, this equality is supposed to be satisfied for all two by two matrix ABCD, that integral matrix. So, ABCD belongs some discrete subgroup of SN2R, right? So, that is modular forms. Of course, there's more growth conditions, one that I don't want to bother you with that. And so, assume that this gamma is the full, the full modular group SN2 of Z, right? And then just apply matrix 1, 1, 1 and C1, 2, 0. You get some, you get F of Z plus 1 had to be going to F of Z, right? So, this one has to be periodic when you move one to the left to the right. And because periodic, you can apply fully, you have to have full expansion. So, FZ had to be of this form. A, N, exponential 2. So, this is clearly invalid when you add 1 to Z, right? It doesn't change. And because of that, you have, this series had, from head to head, this forms. And it's a declare that you look at ANS, interesting arithmetic information, and you put all of them into some DECLAS series. ANN to the penises. So, we, I mean, it's not very clear at the first side why you should do that. But what HECA prove is that this function, this has very much the same properties as the Riemann data functions. That means it has, has Meromorphic continuation and so on. And it has, has, maybe has born or no born dependent with form is Cuspid or not, but. So, now let me move into some, some further properties of the, on this Zeta function and L functions. So, what is very important in the signal now by Riemann himself is not only the data functions that this had Meromorphic continuation, but it has some kind of symmetries, very multiple symmetries. So, in the case of data function, you have to add some kind of mysterious factors, you know, this guy will gamma factors. You add gamma, the user gamma functions and there's some funny numbers, some expression. And then this is C function is symmetrical by one half. So, this is why this line one half is so important because that is symmetric on axis, symmetry axis of the, and this have C of S equal to C of 1 minus S. So, it was easier for me to explain why Meromorphic continuation was important because you want to prove pi number theorem, but now it's more difficult to explain why function equation is important, but it is really the same package and it's very important. And what happens is on other examples, the DLKL functions and HLKL functions attach modular forms, and satisfy the same kind of properties. That means you can have Meromorphic continuation. And if you add some funny factors, then it has some kind of symmetries. Like the DLKL functions satisfy exactly the same equations with some gamma factors, but also some factors depending on some pi p because you know, in the DLKL, you have some prime p that divides the conductors. So, you need to add in some factors depending on those prime p as well as gamma factors at the infinities. In the case of modular forms, you only need to add some gamma factors at the infinities, but also on some factors depending on the primes, depending on your group gamma. So, if you have an SN2Z, you don't need to add anything at finite primes, but if gamma is smaller, then you need to add something. At that stage, it's very confusing what you need to add. It's like a case by case, you need to add something so that you have functional equations. So, I think the real progress was achieved in the interest of add-ins and set out in the data list in the 1950s. So, what Ted does is, he's come by this notion of very... He's quite busy looking at the notion of add-ins. So, add-ins was invented first by Chevrolet and then by Arctin in the classical class field theory. So, when you start with the field of rational numbers, you have different kind of completions. You can do analysis, you need to complete. So, you can complete Q by the usual norms and then get your numbers. And to complete Q by the periodic norms, depends how... Not closed in the usual sense, but closed in the sense that they are different. It's divisible by high power P. And then you complete that way, you get some periodic numbers. And what happens is, you can prove that those are the only possible way to complete rational numbers in some sensible way. And the add-in is I'm quite just put... Oh, I forgot some comma here. You know, you want to multiply all of them. So, put all these big things together. And here I missed a comma because if you don't, you cannot do analysis with this. It's not locally compact. If you put a comma in the rest product, then it's locally compact. And then you can do puncture into all of these and do Fourier analysis. So, even this looks very big, but it's still amenable to do Fourier analysis on this. And what Ted does is he proved that on this construction of Riemann data function, also the DLK function, that seems to come out from analysis. Some numbers can be put entirely in the framework of Fourier analysis. So, all these functions are just some kind of intrinsic function that you would see when you do Fourier analysis. Both on R on QP and also on the whole things, on the add-ins. And so that is why, so first of all, all these factors, 1 minus P minus S minus 1 comes out when you do Fourier and Melanchon form on the periodic numbers. And because you missed, in doing so, you missed the real numbers. If you do the same process over the real numbers, you do get these factors. So, you get these visa-looking factors just by applying the same recipe instead of periodic over the real numbers. So, these whole things is actually some kind of course, you know, not structures, coming structure of Fourier analysis over the add-ins. And then the function equations come out from Poisson summation formula. And I'm not going to say what it is, but when you have locally compact groups, and you have some discrete co-compact subgroups. So, it doesn't, you know, you don't write that because I was going to, you know, it is a fact that Q is locally compact subgroup A. So, the way you understand this, you know, you would say that A is some kind of andrish version of R and Q is andrish version of Z. And so Q inside A is very much like R inside Z, right? So, Z is discrete and co-compact subgroup of R. So, as Q is discrete co-compact subgroup A. And you can make it much more precise. You can see that A mod Q is actually some kind of universal covering of the circle. Right? So, that is one of the best way to understand add-ins. You look at the circles and look at the universal covering, the pro-covering of the circles, and that will be A over Q. And so, you do Fourier analysis on R and on QP and do a million transform. That means you do them on the multiplicative groups. And discrepancy, you know, apply. So, you start with some function, some class of function. So, you can apply the million transform to one side and apply Fourier transform, apply million transform on another side. And you have ratio. And that ratio does depend on the trial test function. The ratio is always the same. And that is the local factors of the pre-magnet data function and the Eclipsial function. And when you, that could be a small and gamma function. And because of the Poisson summation formula, when you multiply all these local factors, you get one. And that is equivalent to the Poisson summation formula. So, that is to say that on this function, that seems to, you know, that you, that demand construct Euler to address some specific problem in number 30, have complete intrinsic existence in Fourier theory. You just term out Fourier theory. Okay. And now the, what happened is the, now it, the, the, the, the HECA story of L function of modular forms also can be completely put into this vision somehow. So, just move from, instead of Gn1, you move to Gn2. And that is, would give you the theory of HECA. So, that was Jacques K. Landlens' book in 1974. So, they just, you know, review the whole HECA theory as automo, L function attached automorphous forms over, over Gn2 of the add-ons. So, now I mean, I cannot explain what it is because it goes very far off from the, from usual knowledge of mathematics, but as it is. And earlier in 1969, 1970, Landlens envisioned a whole story, how it should work. So, it actually, the automorphous form should be attached to some reductive groups. So, reductive group has, any of the usual groups that you know, GnLan, SNLan, orthogonal, symplectic group and so on. It's a couple of other exceptional groups. And the, you would get L function by two data. One is automorphous for the type of G. And the other data is some kind of finite dimension representation of LG. LG is the Condu on groups. That also sometimes appears on physics. In the context, it appears in representation of free groups. And LLan defies L function in the same way as Riemann. That means you just look at the local factors like this one to the p minus s to the minus one. You can have a similar local factors for any pi and for any row. And you multiply them up together. And you can prove that it does convert on some right half plane. And so the whole question is why this function that is well-defined, holomorphic, can be metamorphically continued and have some functional equations. And it's very important known case due to Godman's jacquet. That is given to GnLan. And GnLan and then Duhangubi Gn itself, it has standard representation. And that can go to Godman's jacquet. And Godman's jacquet theory is real. I mean, it's not straightforward, but it's a directionalization of dead series. So it's used by the same technique as dead series. And they do full-edge and form on the space of matrices. And so one of the main goal in theory, is to prove you can found the right to conjectures. It just can be in some kind of a poor man version. You can say that this on-automotive L function are the standard L function. So those that define the word monjoc can standard L function because the row is standard representation. And GnLan's conjecture that is on-automotive L functions should be the same as standard L functions. So that means that you can have possibly a trend first. Or automorphic from on G to automorphic from on GnLan. So that keeps the same L functions. So that is the content of, you know, the one of the big threads in the automorphic forms. And so that's one of the things that I have been working on. In some case, on endoscopic case, you can do this whole transfer problem. It's almost done. And the fundamental lemma is a part of it. And now we, people in the field work on the further case when the transfer is no longer on endoscopic. So, I mean, so we are trying to, in particular, try to generalize this method, good monjoc, from standard into any automorphic representation. It's quite challenging, but maybe it's doable. Okay, so that is one of the phase set of LLan's contenture. That you have automorphic form on reductive groups and have a general notion of L functions. And the basic contenture is the frontorical contenture that you don't get more. The standard L functions are on the L form you can get. And so the other first set is on the reciprocity contentures. So that I, that's kind of L form come from completely different aspect mathematics when has very tough functions. So when you have algebraic varieties, the variable number of things and a set of equations, polynomial equations, you can have algebraic varieties and you count number of points of the algebraic varieties or count number of solutions, modulo p for different p, and you get some numbers and put these numbers in some consistent way and make the product that give you the has-way data functions. And has-way data functions support the same properties as the arrangement data functions. But because LLan, I mean, at least in his philosophy, say that on the L form you get automorphic L functions. So that means on these L functions have to be connected to automorphic form in some way, in pieces of the L function. And that is some of the only way you can prove that has-way data functions can be metamorphic continue and have functional equations. So there's one, of course, a very famous case is known for elliptic curve. I mean, just one single cubic equations, count number of points on the elliptic curve, and you can write down what is has-way data function elliptic curve, and that had to be L function attached to a motor form of way 2. And that is huge achievement of Sir Andrew Wise and his collaborators. And that is done, elliptic curve of a Q. That is a known story now. But what we hope, respect it, it must allow us, not just elliptic curve, not just cubic equation, but any set of equations should have the same properties. But we are, I think, way off from that goal. So now you see the two, really two different facet of the story. One facet is very much more harmonic analysis, you know, or league groups. The structure of the productive group and harmonic analysis on periodic group and real group and complex groups. But that is not so much connected to number theory, probably speaking, but that give you what the L function should be, you know, how you construct a whole new term and how they depend from, you know, how can, what the relation between L functions should be encoded in entirely in the theory of automorphic forms. And application to number theory should go through the reciprocal sequence. I mean, you have set up different equations and have the limited functions, has-way data functions, and they have to be by reciprocating, but you can just connect it to some automorphic L function, right? But then to some standard L function. So you would say that the two connected are quite independent, right? Because after all, you only care about, you can, you know, if you are number 30, you can only care about the standard L functions, because, you know, a posterior, you know that all L functions should be standard L functions. But in practice, it never goes that way, you know, when you prove reciprocal sequence, you need to bring the two connected together, you know, to have jump from one group to other to, you know, to prove some kind of reciprocal sequence. So in doing number theory, now they really have to, very much to grasp what you can do in deformation, very deformation theory in one side, and the known case in frontoriality other side, and she would have, you know, to combine together and get more and more result on the way, because nothing is known in competitive generalities. Okay, that is the content of my talk. I hope I give you some sense about what this general data function is. Thank you. Thank you very much for your very interesting and I think also challenging talk. To understand, I would like to ask for questions from the audience. Yeah. You mentioned that the fundamental, fundamental lemma somewhere appears. Where does it appear? It appears here in frontoriality conditions. It can improve some kind of frontoriality conditions. And it should be used to prove, to go further. So for example, you can transfer automorphic form, SO2M to GN2M, for example, or S in some, so on, it was used by Jim Arthur to prove that the form of automorphic forms on classical group can be transferred to GNL. And for the frontorialities, it's for some kind of very small role, for role like embedding of SO2M to GN2M, for example. But if you do some kind of very big representation like the symmetric to the power 100, then fundamental lemma is not enough. But it's supposed to be in firm ground to build something up on to go to the modern-day case. So reciprocity in algebraic number theory usually has a different connotation. Right? So does that have anything to do with the nomenclature here for reciprocity conjecture? Sorry? Reciprocity usually, when you talk about reciprocity in algebraic number theory, it's related to splitting of primes in algebraic number fields and so on. And here you have, this is also called reciprocity conjecture. Is it just an accident that the reciprocity is used? I think it's the same reciprocity. It is? So how is it the same reciprocity? For example, in the case of quadratic reciprocity, right, you have a quadratic extension of Q, right? And then you look at when the prime P is either split or in note. And that can be detected by a direct character, right? And let's say that one dimension, the representation of the Galov group correspond to the quadratic extensions, correspond to some automorphic form that is directly characters. So I'm just wondering, this work on by Langlands on the groups, is there any generalization to half-algebra? Two? Two half-algebra? Yeah. Half-algebra. Oh, I think there is something that called quantum geometric Langlands program, but I'm afraid not especially that subject, but there is something that called quantum. And actually it is quite, not only people working on it, but I think it gets some real outputs into geometric Langlands, for example. I think then it gets really, it makes a lot of progress on geometric Langlands, and some of the ideas are borrowed from quantum geometric Langlands, and with Hofe and Scheper do play a role that is as much as can say. So I don't see any more questions. I would like to thank Nuba again for his next talk.
 So please come in, have a seat, so that we can start our second part of the morning.  My name is Anna Wienhard, I'm a professor here at Heidelberg University and a member of the scientific committee.  And it's a great pleasure to introduce our next speaker, Nobel Chow from the University of Chicago.  As, I mean, all our laureates, of course he won the FIATS medal as he has won several other awards.  I think he has something which is very untypical even among the FIATS medalists, so his proof of the fundamental lemma made it into the Time magazine's 10 discoveries of the year in 2009, the year before he was awarded the FIATS medal.  And I remember being at lunch at the Institute for Advanced Study when Peter Sarnak, a permanent member in the School of Mathematics, came to the lunch table, very excited because he had just received a call from the Time magazine asking him what he would think of being very important discoveries in mathematics.  And he said, well he told them about the fundamental lemma and he really hopes that it will make it into the Time magazine. And so it was number seven of the 10 discoveries of the year 2009.  And I think the fundamental lemma of course fascinated people in mathematics but apparently also outside of mathematics.  Today, from Bautro, we'll give us another introduction and talk about another fascinating topic, the remensate of function and its generalization. Thank you.  Thank you very much for the introduction. It's a pleasure to have you back here and to give this talk on remensate functions.  I'm not going to talk about fundamental lemma, although at some point I'll say what way it can be used for at some point in my talk. So my talk about this remand data functions, the remand wrote one single paper in number theory, but that is probably the paper who had the deepest influence in the whole development of this theory.  And I try to convey a sense why number theories are so educated by this quite fascinating object that is remand data functions.  But let's step back a little bit into history. This question that has been addressed as early as the Greek times in Orchid. Orchid made this extraordinary proof that there are many prime numbers that I guess one of you know by contradictions.  I'm not going to repeat that proof. Rather I try to explain in a non-regulatory way another proof that due to Euler that he's somewhat less known. So Euler considered this series, you know, sum of n to the minus s as a function of the real variable s.  Of course this series only converges for real numbers as greater than one. It diverges as to how many series diverge.  But what's really interesting is it had this factorization. On the left hand side is a series on the right hand side is infinite product over the prime numbers.  You have this expression because of the unique factorization of integers. Every integer can be factorized in a unique way as for the primes.  And that fundamental property of arithmetic can be encapsulated in this single identity. So it can be an analytic expression of that fundamental theorem arithmetic.  So of course this bold side makes sense only for s greater than one. But now look at what happened at s equal to one. On the left hand side you get the divergent series.  And the right hand side it will be convergent if the set of prime numbers is finite. Every single factor has no point that s equal to one.  So it has only finite product and there will be a problem. So that was the observation of Euler. So that is Euler proof. So that is some of this long series.  So this is the data function. It's really data functions. But Riemann instead of considering the function of real variables, Riemann considers this as a form of complex variables.  And this stuff brings in a huge chain of landscape. So now you have this series, this functions. It converges if the real impact of s is greater than one.  And of course this can be proven easily that it is absolutely convergent uniformly on compact time and so on. So this series, this whole series converges to very nice functions, holomorphic functions on the domain of real impact of s greater than one.  And moreover the Euler arguments in the plies. That means these data functions on this domain have this factorization in product of factors indexed by prime numbers.  So this argument of Euler extends to the complex domain. And so what Riemann is, he proved that this function now can be extended on the whole domain of complex plane.  But it catch poles at s1 to 1. So it's the only single pole at s1 to 1. And in that way he can make sense of these kind of funny identities when you put s1 to 0.  So of course this doesn't make sense. This is 1 plus 1 plus 1. But then if you compute the value of this function, it's not completely fine. And you get zeta of 0, you can 1 minus 1 half. And sometimes 1 plus 1 plus 1 is minus 1 half in Riemann's sense.  And if you put s1 to minus 1, it's even more funny. So 1 plus 2 plus 3 would be minus 1, 12. So that is... Anyway.  So these papers have had a huge influence in number theory because it set a way into understanding prime number theorem. So how many are their primes less than given magnitude?  So the argument goes as follows. Here you have zeta. Zeta is a product over primes. If you take a logarithmic derivative, right? And then product becomes sum.  So you get... This is a logarithmic derivative of zeta. And you can develop into a sum. And what happens is it's almost a sum over prime numbers.  So this lambda n is called von Machen functions. Lambda n is a log d if n is a power of prime numbers. So if lambda of 60 is 0, for example. So basically when you take a logarithmic derivative, write...  You have written it as series. You only get... Only left the coefficient corresponding to prime numbers. This is actually sum over primes.  So if you didn't know pi x, the number of primes less than given real number x, then the prime number theorem that was actually already suspected. I mean, hypothesized by Gauss and Legendre by numeric calculations.  Pi x should be equivalent to x over log x. x has to go to infinity. And pi x is basically a sum of 1 plus 1 over the set of prime numbers.  Now, because there's not so much difference between the prime and the set of power primes, it's not... It's quite elementary to prove that this...  The prime number theorem is equivalent to this. So instead of counting prime, we just sum the von Machen function up to n less than given x.  And if it's equal to say this sum is equivalent to x, then x goes to infinity. So this requires an argument, but not...  And we talk to you with those arguments, but it's very elementary. So now we see the relation between the prime number theorem with von Machen functions and with Lorentz derivative data functions.  So this is... When you look at Lorentz derivative as the series, you get this lambda n as coefficient of the direct-clase series. And now the question is how to estimate this sum of lambda n.  Okay. So now we come a little bit of harmonic analysis. So let's look at some questions on the direct-clase series. Let me just sum n to the s with coefficient of some given function.  You can be von Machen function can be anything. And with some... Now, you really see why it helps to look at s as complex parameters, because it can use complex integrations.  Instead of having only real and lie, you can integrate your vertical lines. And so now if you get this function, you integrate it over some vertical line.  So let's assume that this series is converted absolutely, right? When s... When a real and particular s greater than sigma zero, some numbers.  So we have nice convergent series on some half planes. And on that half plane, you can estimate this sum, right? a n, this part sum sum, a n, n less than x by these co-integrals of vertical lines.  So it looks a little bit complicated, you know, but the very basic things, you know, if you don't... if you only care about x.  So the only occurrence of x is actually the s, right? So if you ignore all the difficulties, we'll say that this is basically actually the s, because that is the only thing that counts, right?  So you have estimate, you can estimate the sum of the left by some constant time, actually the sigma. So that's, you know, very roughly the basic information you can get from this complex integration.  But now this should not depend on sigma, right? Because it... So you get better estimate by moving your line integration.  As long as the line integration is contained in the convergent domain, there's no difference doing that. And you can move in integration on the way where you get some points, right?  So I mean, you get some points, I mean, in comparison, it's not a problem, just move across a point and you catch the residues, right? And just do... doing so by doing... by this trick of... this is very basic tricking as number theory, just moving the line integration further to the left as much as you can, right?  And that you can get some kind of asymptotic expression of this sum, right? The main term should be the residues that you catch along the way moving your line integrations, right?  And you can do it as much as you can analytically, and continue your complex function. So that's why it's very important to get the meromorphic continuation of the data function, right?  And when you get it, you can also meromorphic continuation of this locality derivative and so on.  Okay, so now we want to understand this... to apply this trick, this complex integration to this... the locality derivative of data, right?  And so we need to understand, you know, anyone... then you say the lambda 1 plus lambda 2 plus etcetera plus lambda n can be estimated by the points of the locality derivative of data.  But the point of this is exactly the points and the zeros of data. So that is why you really need... data doesn't have... to have one point, it's very simple.  To put it one. So what do you call as zero data function? So that's why it's such a big deal to understand zero data functions.  So for example, by following this... by the strategist Adam M. M. de la Valle Poussin, prove rigorously at the beginning of the 20th century that there is some... you can extend slightly over the line...  So I have a real property of s1 to 1, so I have some zero free zone. You know, you get closer and closer to 1 when the imaginary part gets to infinity, but it's really the open part.  And just with this, they can prove the pi number theorem. Yes, lambda 1 plus lambda 2 plus lambda n is equivalent to n.  But of course the demand condition is much stronger. He's... the zeros are on the critical line of real property of s1 to 1,5.  And somehow the estimate can never go past that line. That is the best you can do, which is go closer, closer to real property of s1 to 1,5.  Okay. So that I tried to say in very rigorous way ideas in demand papers and the development further.  Of course there's a lot of difficulties, I ignore all the possible energy difficulties in doing this, you know, inverse Fourier transform and so on.  But now let's... one of the... I think it's very remarkable if this storage doesn't apply only to Riemann data function, but to many similar functions.  So one of the basic examples that come on mode at the same time as Riemann is this Conductor L function. So it's very similar.  But we have some more data. So you start with some n, Conductors is natural numbers.  And you are giving a characters of the multiplicative group Z and Z to C star. So let's pick an example at n or 2,3.  And so all we have one, two class that is... so Z mod 3 has three classes, modulo 3. And the cross just delete this zero.  So the characters can pick one when n is going to 1 modulo 3 and minus one when n is going to 2 mod 3.  And so form this function on the class from the modulo n, you can just extend it to Z, just put in zero where the function is not defined earlier.  And then you can just form this series. So it's completely similar to the Riemann data functions, but instead of one, you put one or minus one, you know, or unit roots, depending on your characters.  And of course, if n, if here, big n, 1, 2, 3, then there's no more term with n, 1, 2, 3. So term is just left out.  And very similar to Riemann data function, you can improve this function. It has metamorphic continuation.  And actually it's more, quite amazing. It has no bones. If the guy is not trivon characters, there's no bones.  And that is how directly prove that there's infinite primes in arithmetic progressions.  So that is, you know, you can count the number of prime, you know, arithmetic progressions by the same kind of tricks, you know, and just apply the Perron, this is called the Perron formula, to this directly L functions.  And you get the into that. There's no, and moreover, they don't have no bones. And also, it is, this L function is supposed to satisfy Riemann hypothesis also on this interesting zeroes,  must be on the lie of Riemann data, one half. But of course, nobody have a single clue how it's true.  Okay. Now, let me move on further. So as in, as was emphasized in S. and Riemann lectures, number three had to move at some point from a billion to non-nibbillion.  And here the, the break point is the theory of, or HECA theory of L form attached to modular forms.  So, I mean, I haven't studied enough of history to know, but I guess this, that I was not clear why it is connected, even connected to Riemann data function, although it has many similar properties.  So, history is called modular forms. Modular form is essentially some kind of whole-on-mode function on the, on the half plane, right?  And this satisfies a modular relation. So, like, AZ plus B over CZ plus D is, it doesn't change the function up to some, some factors.  So, this K is the weight, the weight of the modular forms. And this function, this equality is supposed to be satisfied for all two by two matrix ABCD, that integral matrix.  So, ABCD belongs some discrete subgroup of SN2R, right? So, that is modular forms. Of course, there's more growth conditions, one that I don't want to bother you with that.  And so, assume that this gamma is the full, the full modular group SN2 of Z, right? And then just apply matrix 1, 1, 1 and C1, 2, 0.  You get some, you get F of Z plus 1 had to be going to F of Z, right? So, this one has to be periodic when you move one to the left to the right.  And because periodic, you can apply fully, you have to have full expansion. So, FZ had to be of this form.  A, N, exponential 2. So, this is clearly invalid when you add 1 to Z, right? It doesn't change.  And because of that, you have, this series had, from head to head, this forms.  And it's a declare that you look at ANS, interesting arithmetic information, and you put all of them into some DECLAS series.  ANN to the penises. So, we, I mean, it's not very clear at the first side why you should do that.  But what HECA prove is that this function, this has very much the same properties as the Riemann data functions.  That means it has, has Meromorphic continuation and so on.  And it has, has, maybe has born or no born dependent with form is Cuspid or not, but.  So, now let me move into some, some further properties of the, on this Zeta function and L functions.  So, what is very important in the signal now by Riemann himself is not only the data functions that this had Meromorphic continuation,  but it has some kind of symmetries, very multiple symmetries.  So, in the case of data function, you have to add some kind of mysterious factors, you know, this guy will gamma factors.  You add gamma, the user gamma functions and there's some funny numbers, some expression.  And then this is C function is symmetrical by one half.  So, this is why this line one half is so important because that is symmetric on axis, symmetry axis of the, and this have C of S equal to C of 1 minus S.  So, it was easier for me to explain why Meromorphic continuation was important because you want to prove pi number theorem,  but now it's more difficult to explain why function equation is important, but it is really the same package and it's very important.  And what happens is on other examples, the DLKL functions and HLKL functions attach modular forms,  and satisfy the same kind of properties. That means you can have Meromorphic continuation.  And if you add some funny factors, then it has some kind of symmetries.  Like the DLKL functions satisfy exactly the same equations with some gamma factors,  but also some factors depending on some pi p because you know, in the DLKL,  you have some prime p that divides the conductors.  So, you need to add in some factors depending on those prime p as well as gamma factors at the infinities.  In the case of modular forms, you only need to add some gamma factors at the infinities,  but also on some factors depending on the primes, depending on your group gamma.  So, if you have an SN2Z, you don't need to add anything at finite primes,  but if gamma is smaller, then you need to add something.  At that stage, it's very confusing what you need to add.  It's like a case by case, you need to add something so that you have functional equations.  So, I think the real progress was achieved in the interest of add-ins  and set out in the data list in the 1950s.  So, what Ted does is, he's come by this notion of very...  He's quite busy looking at the notion of add-ins.  So, add-ins was invented first by Chevrolet and then by Arctin  in the classical class field theory.  So, when you start with the field of rational numbers, you have different kind of completions.  You can do analysis, you need to complete.  So, you can complete Q by the usual norms and then get your numbers.  And to complete Q by the periodic norms,  depends how...  Not closed in the usual sense, but closed in the sense that they are different.  It's divisible by high power P.  And then you complete that way, you get some periodic numbers.  And what happens is, you can prove that those are the only possible way to complete rational numbers in some sensible way.  And the add-in is I'm quite just put...  Oh, I forgot some comma here.  You know, you want to multiply all of them.  So, put all these big things together.  And here I missed a comma because if you don't, you cannot do analysis with this.  It's not locally compact.  If you put a comma in the rest product, then it's locally compact.  And then you can do puncture into all of these and do Fourier analysis.  So, even this looks very big, but it's still amenable to do Fourier analysis on this.  And what Ted does is he proved that on this construction of Riemann data function,  also the DLK function, that seems to come out from analysis.  Some numbers can be put entirely in the framework of Fourier analysis.  So, all these functions are just some kind of intrinsic function that you would see when you do Fourier analysis.  Both on R on QP and also on the whole things, on the add-ins.  And so that is why, so first of all, all these factors, 1 minus P minus S minus 1 comes out when you do Fourier and  Melanchon form on the periodic numbers.  And because you missed, in doing so, you missed the real numbers.  If you do the same process over the real numbers, you do get these factors.  So, you get these visa-looking factors just by applying the same recipe instead of periodic over the real numbers.  So, these whole things is actually some kind of course, you know, not structures,  coming structure of Fourier analysis over the add-ins.  And then the function equations come out from Poisson summation formula.  And I'm not going to say what it is, but when you have locally compact groups,  and you have some discrete co-compact subgroups.  So, it doesn't, you know, you don't write that because I was going to, you know,  it is a fact that Q is locally compact subgroup A.  So, the way you understand this, you know, you would say that A is some kind of  andrish version of R and Q is andrish version of Z.  And so Q inside A is very much like R inside Z, right?  So, Z is discrete and co-compact subgroup of R.  So, as Q is discrete co-compact subgroup A.  And you can make it much more precise.  You can see that A mod Q is actually some kind of universal covering of the circle.  Right? So, that is one of the best way to understand add-ins.  You look at the circles and look at the universal covering,  the pro-covering of the circles, and that will be A over Q.  And so, you do Fourier analysis on R and on QP and do a million transform.  That means you do them on the multiplicative groups.  And discrepancy, you know, apply.  So, you start with some function, some class of function.  So, you can apply the million transform to one side and apply Fourier transform,  apply million transform on another side.  And you have ratio.  And that ratio does depend on the trial test function.  The ratio is always the same.  And that is the local factors of the pre-magnet data function and the Eclipsial function.  And when you, that could be a small and gamma function.  And because of the Poisson summation formula,  when you multiply all these local factors, you get one.  And that is equivalent to the Poisson summation formula.  So, that is to say that on this function, that seems to, you know,  that you, that demand construct Euler to address some specific problem in number 30,  have complete intrinsic existence in Fourier theory.  You just term out Fourier theory.  Okay.  And now the, what happened is the, now it,  the, the, the, the HECA story of L function of modular forms also can be completely put into this vision somehow.  So, just move from, instead of Gn1, you move to Gn2.  And that is, would give you the theory of HECA.  So, that was Jacques K. Landlens' book in 1974.  So, they just, you know, review the whole HECA theory as automo,  L function attached automorphous forms over, over Gn2 of the add-ons.  So, now I mean, I cannot explain what it is because it goes very far off from the,  from usual knowledge of mathematics, but as it is.  And earlier in 1969, 1970, Landlens envisioned a whole story, how it should work.  So, it actually, the automorphous form should be attached to some reductive groups.  So, reductive group has, any of the usual groups that you know,  GnLan, SNLan, orthogonal, symplectic group and so on.  It's a couple of other exceptional groups.  And the, you would get L function by two data.  One is automorphous for the type of G.  And the other data is some kind of finite dimension representation of LG.  LG is the Condu on groups.  That also sometimes appears on physics.  In the context, it appears in representation of free groups.  And LLan defies L function in the same way as Riemann.  That means you just look at the local factors like this one to the p minus s to the minus one.  You can have a similar local factors for any pi and for any row.  And you multiply them up together.  And you can prove that it does convert on some right half plane.  And so the whole question is why this function that is well-defined, holomorphic,  can be metamorphically continued and have some functional equations.  And it's very important known case due to Godman's jacquet.  That is given to GnLan.  And GnLan and then Duhangubi Gn itself, it has standard representation.  And that can go to Godman's jacquet.  And Godman's jacquet theory is real.  I mean, it's not straightforward, but it's a directionalization of dead series.  So it's used by the same technique as dead series.  And they do full-edge and form on the space of matrices.  And so one of the main goal in theory,  is to prove you can found the right to conjectures.  It just can be in some kind of a poor man version.  You can say that this on-automotive L function are the standard L function.  So those that define the word monjoc can standard L function  because the row is standard representation.  And GnLan's conjecture that is on-automotive L functions  should be the same as standard L functions.  So that means that you can have possibly a trend first.  Or automorphic from on G to automorphic from on GnLan.  So that keeps the same L functions.  So that is the content of, you know, the one of the big threads in the automorphic forms.  And so that's one of the things that I have been working on.  In some case, on endoscopic case, you can do this whole transfer problem.  It's almost done.  And the fundamental lemma is a part of it.  And now we, people in the field work on the further case  when the transfer is no longer on endoscopic.  So, I mean, so we are trying to, in particular,  try to generalize this method, good monjoc,  from standard into any automorphic representation.  It's quite challenging, but maybe it's doable.  Okay, so that is one of the phase set of LLan's contenture.  That you have automorphic form on reductive groups  and have a general notion of L functions.  And the basic contenture is the frontorical contenture that you don't get more.  The standard L functions are on the L form you can get.  And so the other first set is on the reciprocity contentures.  So that I, that's kind of L form come from completely different aspect  mathematics when has very tough functions.  So when you have algebraic varieties,  the variable number of things and a set of equations,  polynomial equations, you can have algebraic varieties  and you count number of points of the algebraic varieties  or count number of solutions, modulo p for different p,  and you get some numbers and put these numbers in some consistent way  and make the product that give you the has-way data functions.  And has-way data functions support the same properties as the arrangement data functions.  But because LLan, I mean, at least in his philosophy,  say that on the L form you get automorphic L functions.  So that means on these L functions have to be connected to automorphic form  in some way, in pieces of the L function.  And that is some of the only way you can prove that has-way data functions  can be metamorphic continue and have functional equations.  So there's one, of course, a very famous case is known for elliptic curve.  I mean, just one single cubic equations,  count number of points on the elliptic curve,  and you can write down what is has-way data function elliptic curve,  and that had to be L function attached to a motor form of way 2.  And that is huge achievement of Sir Andrew Wise and his collaborators.  And that is done, elliptic curve of a Q.  That is a known story now.  But what we hope, respect it, it must allow us,  not just elliptic curve, not just cubic equation,  but any set of equations should have the same properties.  But we are, I think, way off from that goal.  So now you see the two, really two different facet of the story.  One facet is very much more harmonic analysis, you know, or league groups.  The structure of the productive group and harmonic analysis on periodic group  and real group and complex groups.  But that is not so much connected to number theory, probably speaking,  but that give you what the L function should be, you know,  how you construct a whole new term and how they depend from, you know,  how can, what the relation between L functions should be encoded  in entirely in the theory of automorphic forms.  And application to number theory should go through the reciprocal sequence.  I mean, you have set up different equations and have the limited functions,  has-way data functions, and they have to be by reciprocating,  but you can just connect it to some automorphic L function, right?  But then to some standard L function.  So you would say that the two connected are quite independent, right?  Because after all, you only care about, you can, you know,  if you are number 30, you can only care about the standard L functions,  because, you know, a posterior, you know that all L functions should be  standard L functions. But in practice, it never goes that way, you know,  when you prove reciprocal sequence, you need to bring the two connected together,  you know, to have jump from one group to other to, you know, to prove some kind of reciprocal sequence.  So in doing number theory, now they really have to, very much to grasp  what you can do in deformation, very deformation theory in one side,  and the known case in frontoriality other side, and she would have,  you know, to combine together and get more and more result on the way,  because nothing is known in competitive generalities.  Okay, that is the content of my talk.  I hope I give you some sense about what this general data function is.  Thank you.  Thank you very much for your very interesting and I think also  challenging talk.  To understand, I would like to ask for questions from the audience.  Yeah.  You mentioned that the fundamental, fundamental lemma somewhere appears.  Where does it appear?  It appears here in frontoriality conditions.  It can improve some kind of frontoriality conditions.  And it should be used to prove, to go further.  So for example, you can transfer automorphic form, SO2M to GN2M, for example,  or S in some, so on, it was used by Jim Arthur to prove  that the form of automorphic forms on classical group can be transferred to GNL.  And for the frontorialities, it's for some kind of very small role,  for role like embedding of SO2M to GN2M, for example.  But if you do some kind of very big representation like the symmetric to the power 100,  then fundamental lemma is not enough.  But it's supposed to be in firm ground to build something up on to go to the modern-day case.  So reciprocity in algebraic number theory usually has a different connotation.  Right?  So does that have anything to do with the nomenclature here for reciprocity conjecture?  Sorry?  Reciprocity usually, when you talk about reciprocity in algebraic number theory,  it's related to splitting of primes in algebraic number fields and so on.  And here you have, this is also called reciprocity conjecture.  Is it just an accident that the reciprocity is used?  I think it's the same reciprocity.  It is?  So how is it the same reciprocity?  For example, in the case of quadratic reciprocity, right,  you have a quadratic extension of Q, right?  And then you look at when the prime P is either split or in note.  And that can be detected by a direct character, right?  And let's say that one dimension, the representation of the Galov group  correspond to the quadratic extensions, correspond to some automorphic form  that is directly characters.  So I'm just wondering, this work on by Langlands on the groups,  is there any generalization to half-algebra?  Two?  Two half-algebra?  Yeah.  Half-algebra.  Oh, I think there is something that called quantum geometric Langlands program,  but I'm afraid not especially that subject, but there is something that called quantum.  And actually it is quite, not only people working on it,  but I think it gets some real outputs into geometric Langlands, for example.  I think then it gets really, it makes a lot of progress on geometric Langlands,  and some of the ideas are borrowed from quantum geometric Langlands,  and with Hofe and Scheper do play a role that is as much as can say.  So I don't see any more questions.  I would like to thank Nuba again for his next talk.
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