 Okay, it's recording. We'll share this thing. But this one, share. Okay. Should be there. But there. Is that okay? Okay. Okay. Okay. So if you like one of these two guys, it's probably going to be. Okay. And she was a former student of the manner. So we know her very well. So. Sorry. Thank you for the introduction, but the same. I'm a lot of people who didn't meet here again very happy. So. And I'm going to introduce today to introduce some of the topics that I have worked with that. But years in the HD and also in the best. So if the peanuts Kristen said I'm going to talk about some topics of the interface of analytic numbers theory and harmonic analysis, such as career analysis. So. Today's story. This is the career. So when a function. We can defy that in this way. And this is very useful. And this comes out of the time you're on the side of the matter. For example, in when you have electronic signals, like to make a call from a cell phone, you want to transmit the signal of this phone call. And you want to recover it in a precise way. So you want to transmit on certain frequency. So here, if you have a function F, you define the field transform as the integral with the function F. And this factor is to the minus two part I X. So business. I thought that financial so it is alternating in that unit circle and stuff where we have the population. Of course we can grab. We can write an exponential that. In terms of science and health. So. And one big question in the field is that for your uncertainty. So the question is up to what points can we post. Some conditions on a function. And on the four year transition will change me. And we want to recover a function and see what's the best possible way to do this. So again, for example, you have some electronic signal to have some restrictions in the frequency. So. And. You can start a problem directly. And there are many different ways for the life of this idea. Like we did it. Or work with them. Well, but it also appears in many for your station problems that are connected to lots of different areas, like numbers, theory of the opportunity. So this was transferred to be our first today. But now we want to talk about. That we want to have a function. So when you have a X greater than one word, you could think of X as a complex number, real part of X greater than one. We can define the function as the sum. And it comes compared to an analytic function, but also we have some over integers. And if we factor the integers by the fundamental theorem of our math, we can write this in terms of this product over the prime numbers. So in the sense, we can think of this as the starting point of analytic numbers theory, because we have a relation between prime numbers on one hand. And now it's an electric function on the other. So by manipulating the presented in terms of the analytic properties of this function, we can recover a lot of information and prime numbers. And if we take a little bit of complex analysis, we can show that we can actually extend it to a function. So the analytic function in the complex plane, just a simple whole of the one. And it satisfies this functional equation. So this gives a relation between the value of S and the value of one minus X. And if we stick, this is the complex plane, if we stick to the right of this line, where real part of X is greater than one. So the first formula holds. And here we know we can write a function as a product over prime product of things that are not zero. So this is not going to be zero here. There are nothing to be a zero in this part. And because of the symmetry, so that time here has some zeros. So this means that the reason for the function is going to have some zeros here at minus two, minus four, minus six, at all of the negative integers. And those are what we call the trivial zeros. But there cannot be any other zero here to the left because of this functional equation and the other parts and a little bit of work. So looking at all of the other materials, both like here in this part between zero and one. And because it's a critical strip, and we can prove this today, but the real hypothesis says that all heroes should actually like here in a little in this line. And because the critical line is the line real part as equals to one half, but this is actually the hard problem for many years. We can define other related functions. So like I said, the function is useful to study prime, but sometimes we are interested in studying prime to some special structure. So for example, for prime arithmetic progress, it is useful to study that there is a cell functions and still they are defined initially in a similar way. But now we have in the numerator, this function kind of 10. And those are the things that character sense, we can think of them as some, this is the most expected superiority function overseas. So this has two and nine properties. Instead of superiority in a sense it captures the structure of arithmetic progression, but also it is more negative. So we still congratulate it as a product of our prime numbers. So now we can use this study to study prime with this special structure. And there's also another one that I like. It's called a heck of a function. And we can think of this as a general educational, there's a function to analyze the right number field. This is like today, the ratio of numbers and you add for example, because the numbers won't read like we added it. So this is a productive number field. And for some, for some of the right number fields with this, we can find this heck of a function in a similar way. So now instead of a sum over a integer, we have a sum over integral ideals. And in the denominator, you have that more of which ideals, which is an integer. And in the numerator, we have this special heck of a character. And we can think of them as a completely multiplicative function over integral ideals, multiple sum integral ideals here. And in the same way that these health functions are useful to study primary and practical reasons, this heck of a function can help us study practical ideals with a special structure. And as we will see this also helps to study primes represented by quadratic points. But now let's go back to the case of dynamic function. We want to count how many heroes there are up to a certain type. So let's go back to this graph. We want to count the heroes inside this region in the critical state from hero to a certain high speed. So this, the number of heroes inside this region is what we call energy. And you think a little bit of complex analysis. The Riemann-Mann matrix formula gives us a very precise way to activate this number of heroes. We first have a continuous part, and we have this special function as a speed and a very good character. We are counting the speed, as here because it's a function, so it's going to have a lot of continuity for the first part to begin with. So this means that part of the energy part of the communities are going to happen in the function as a speed. So if you really want to understand the distribution of the heroes, we want to understand the function s. This function s is defined in this way. Here is the argument of the Riemann-Mann function in that critical line. It can be defined in terms of the logarithmic derivative, for example, and it satisfies some bounds. It is smaller than the p, and it's integral cos of small. So this means that it has a lot of population and constellation. So that means that if we want to understand the distribution of the heroes, we really need to understand the observatory and statistical information of this function s. We also want to obtain some statistical and observatory information of the function s of p. Formula here, because of the Riemann-Mann model formula, we can expect that there should be around delta heroes in this short interval from t to t plus 2.0 over lambda p. So we want to study this with more precision. This is what we call the number variance. This is the variance in the statistical sense of the number of heroes in this interval. So this is the mean square of the number of heroes minus p, expected value. And because of this formula, we can write this as p. So again, to understand the zero coefficient of s. So okay, why are we interested in studying the zero-competent function without the distribution of the heroes is very closely connected to the distribution of s. Like we were in the beginning and this can be seen by using, for example, the so-called explicit formula. And these are the quantities that connect expressions involved in zero with expressions involved in s. So if we understand the zero, we can understand s. And for some questions, to understand the number of heroes, the number of s. The real hypothesis gives us the best possible information that we can hope for. Like this if I'm not mistaken. So for some questions, the real hypothesis gives us the best possible information that we want. And for other questions, like if you want to understand the short interval, we need to go beyond the real hypothesis. So we are going to understand the primary vertical distribution of the heroes. The real hypothesis and all of the heroes have been formed without the s. And we want to understand the distribution of this ordinance. To do this, well, by the formula, we can see that the average of the heroes must be too far over the last few. Just by applying this first formula here. So we want to go beyond it. So we are going to try to understand the real hypothesis. If we choose one zero, we want to come and average how many zero we can expect at a certain distance. This is what Montgomery wanted to do. Or a table before a last set function r. We want to study sounds of this type. Formulated as a gesture for the behavior of this. So let's see what this means. Here on the left, we have the account quantity beta and the average gap. So if the heroes were to be random points, if they were to be random points, we would expect for random points that the sum of all pairs of points should be beta times the number of heroes. They are not distributed from random points. We have factors, we have factors that define you or you are giving us a distribution, giving us the this formula. So this is to be written differently. This is what Montgomery said and furthermore Montgomery noticed that this is the same function when it says that the gaps between the agent values of some resonations The ordinates of the non-trivial zero are distributed like the eigenvalues of a random message from a person who already has a solution called the Gaussian Indicator Incentive. And this is a very nice indexer because it fits very nicely with another indexer of purpose and volume, which says that the ordinate of the zeroes must be the eigenvalues of a 13-permitian operator. And if this were true, this would imply the probability positive. So it's a very nice injector that goes beyond the real and positive. Excuse me. What is the ordinate? Like zero, but a pair of A is beta plus beta. So I mean, this, the imaginary part, the whole fashion term for the imaginary part of the company. Can you say to me if you're calling what you used in the correctional idea? Yes. No, no, no. So we mean that the dynamics are just good and it's like to understand how there should be expressions like this. And if we want to do this, then we can apply a little bit for your analysis. So we are going to introduce here a way just to make the problem simpler and a little bit. And we are going to normalize by last year versus five again for to simplify the analysis. And so to understand this out here is very similar to understanding the sum in the top. We just have a way in a normalization. And if we apply for your inversion, then this sum can be written in this way in the right. And here we have the special function F. This was from number something F. We can describe it in this way. So we can think of it as a way in a normalized Fourier transform of the distribution function of the pairs of fields. Because of this formula here, which is just for getting a version, if you want to understand this sum in the top or this sum here in the second slide, we need to understand the right hand side. Here we have the table of F with the Fourier transform of part. So to understand this, we need to understand the behavior of this function F. And this is historically at T goes to infinity. Well, the problem with estimating the sum of what we do is the problem of understanding this. And that's what Montgomery and then Gostin and Montgomery did. They showed that historically when T goes to infinity and we now try it to small in 0 and 1, we have a very precise formula for that. So in other words, when we now try it small, we can understand this expression. But we want to know what's the behavior for all of us. And Montgomery made a suggestion that when outside greater than 1, F is just approximately 1. But this looks like a very hard trajectory and it has been connected to very hard problems in number theory like the behavior of furniture interval for the behavior of people since during the function. So it's still very far outside. And that's what we can do. That is true. Then just by shooting special function R, we can obtain the other version of the fair for relation to lecture. So this formula is called a strong fair for relation to lecture. As far as the function F of T, so what is that to understand the distribution of the theorem? We want to understand the behavior of F. So for example, study the statistical information, the statistical behavior of F of T. And we can use the method of moments into ability. If you want to understand that relative distribution, you can start by studying moments of the business and separate obtained all of the time and moments. Assuming there is a hypothesis in it, it can also be obtained unconditionally with a weaker error. So it's a consequence that we know the moment, we know the probability distribution, this constant in the moment behaves like moments of a thousand. So this means that we have a central limit theory. This, that means that F of T behaves like a thousand. It's distributed like a thousand. We mean zero and with variance square with a one half level of two. So we have a very complete statistical information for the function F, which is what we want. And similarly, so I think that F of T is the imaginary part of the logarithm of the Riemann Center function in the critical length. So we can also study the real parts and to start with, you have to see behavior. And so this is true behavior, like independence, guidance with this mean and back. So the same thing and hope for, for a lot more sense in the critical length. They are both normally distributed with means zero and this burns. And here are some pictures. Here on the left, we have the numerical data distribution of the real part of the logarithm and the imaginary part of the logarithm. So this is F of P and this is plus. And we can see that they both look very close to a Gaussian, but it's interesting that this one looks much closer to a Gaussian than this one. This one's like a little bit skewed. And so far, far as I know, there has to not be any explanation for this unit because without that in the link, they both should converge, but this one converges. So this graph is taken from this paper getting on tonight, but the data was included by us. So we might want to try to understand why there is a divergence in the two cases of the real part of the logarithm and the imaginary part of the logarithm. So for example, we can try to compute the variance with more precision. And it was done by Cauchy in 1988. Assuming the right hypothesis, he estimated the second moment, but having the second order term. So the first on their term is the one from the second one. Now we also have the second one. And for the second one, this is very interesting because this is this constant A of T and this constant contains information from time and effort and also contains information from the fields. So for the information on the fields, it's categorized in this function and this is from the function for better relation that I mentioned before. So this has information on the fields. And we can try to do the same for the real part of the logarithm to see if there's any difference. And we did this recently, but it turns out that the answer is the same. So for me, it is surprising because the proofs are actually a little bit different and it's not obvious that they should get the same answer from the start, but again, it's somehow the same thing. So this is still doesn't explain the difference in the class that we just saw. So we want to go far to understand the difference. And this is working progress between right now. We want to understand this unit in the graph. So the unit of the logarithms is the relation can be captured by the third moment. So we want to understand the third moment of FFT. And it turns out here we do affect the difference. So for FFT, this third moment should be very, very small and this makes sense because this is what we expect for a Gaussian. For a Gaussian, the third moment is just fewer. But for the real part of the logarithm, it is not that small. We expect a term inside a constant times p. So this will explain in some way the difference in the numerical class. And in particular, this is what I have here. It's a special case, but more general structure of getting on stage that they obtain using this model of random matrices. And it's an interesting because we expect to see different results, but it's also much harder because we need information from the third relation of the zero for a Gaussian like I just showed you. But here we will get some information from triple to relation and so far, it is not clear how to proceed, but this is something that we are working on. We've talked a little bit more about the number of variance of the theorems. So I think you just want to see. We want to obtain this special information of the same with that. The virus of the number of few within a certain short interval, and because we're just in terms of as a P up to a very small error. This is the number but inside by many answers such as very kidding foodie, gotta learn very meet a very precise trajectory about the behavior of this one. So first, as we said, there is that hope for the objective that says that the heroes. That your name of the series should be that I can buy some hermit to my career. The nation actually behaves like way back to some restaurant. It turns out that this injection of the government according to some medical evidence, it is very good for some short. Zero that are very close together. But it's failed for long. This is effective between zero that are very far apart. So we want to understand that relations between zero that are very far from each other. And that we need some extra information from prime numbers. This is what is the noise. So that later in 1988, barely introduced a new model for the people. And he uses a few of the items of 13. This is an operator from from dynamical sense and with this model, it actually fits all of them. The short range of model for the universe of a gene and the long range of our cause non-indiverse of a gene and so far works perfectly well with all of them. So that very use this new model for the zeros to conjecture a very precise behavior for the number of hours. This location as I was in their side integral and see I give us in their cosine. So this was very good. This is not for them. The important part is that we have to park. So we have some shit. Little bill. This little outside is the size of the interval that we're trying to find. So it's very small. This is the first part a little. So it's a little small. The conjecture some behavior. And what is interesting about this is that behavior here for a, it's exactly what we will obtain from random ages. And that's the important part. We have a very different behavior. So no longer work in this range. Instead, we need to use prime numbers for a number of captures in this function. This is the function. It contains. We can take this. I guess it's fine frame numbers. This is. The. Our time and you know, I don't know. I don't find prime numbers. And we need this prime number for the second part. So this really has a very different behavior depending on the range. And that's very good. Part of first and then several approaches so far. So for example, Fuji, the first part of the third part is this. Let's know. We don't know what it is called the shift. So with that shift. So that look of response to part eight year with that shift. Then he's able to compute this. And again, for this second term here, we get information from the theorems in the percolation, the function for percolation, introduced by Montgomery steps. And in particular, for this formula, if we also assume Montgomery is a strong fair correlation trajectory, if f is just asymptotically one, then this implies there is a vector in part a, in the fire point, the right is small. But however, we really expect a different behavior in the second part. So we need to use, we really need to use other tools to be able to tackle this second part. So for that, we introduce this function. It is a function introduced by Shan, it is a formula of 4th, it is defined in this way. It is very similar to Montgomery function, but we now have some parameter, big delta here. And it appears here. So instead of counting that creates the pairs of zeros, the difference is very small. We want the difference in a pair of zeros to be close to this quantity delta. It means that the zeros must be very far apart. So this is exactly what we want. We are, this is a tool to study zeros that are very far apart. And if we use this tool, for all the chance of taking a formula for this, now five is small, and similar to Montgomery, it conjectures the behavior without firing large. This is like a transversion of Montgomery's fair correlation trajectory. And using this tool, we were able to obtain a formula for the number of times when the shift is much larger, the load of less frequency. And we can see that here for this second term, we need to use this function f delta apart from Montgomery's function f. So I think the takeaway here is that we really need to use this different tool to be able to obtain a formula that works uniformly for both small delta and big delta because we expect a different behavior in the two range. As a consequence, we feel using this result, if we assume that it can be possible to enchant version of the fair correlation for a delta, then there is a conjecture hold in both range for all delta after this side. So what's interesting here is that for the first time, we have covered the two range, the universal and the non-universal, part A and part B of the conjecture of fair. Let's talk a little bit about integers represented by both other forms. So here we'll start with this classical result of premise. We know that a frame contributed at the sum of two squares, if and only if, it is covered in one month fourth. And actually, for math proof stuff, we need a self, for example, we can ask what happens if a frame is x squared plus 2 1 squared. And if a frame shows that this has a similar characterization, a frame of this fourth is only if it is covered in one or three month eighths. So something very similar, just a union of terms of capital four, a union of mathematical reference. But there are other examples that are more possible for example, historically, this was one of the first ones. If a frame of the fourth x squared plus 7 1 squared, then the characterization is that the frame must become one with three and two must be a cubic residue, must be. So now it's a little more interesting, and this kind of thing we didn't in the previous weeks, it was actually a conjecture of two hundred and it was there for a week. It was there for, like, a year of cubic present present. And to show you that we cannot yet use our mathematical reference to come to this, we can consider this with this other expression, 4 x squared plus 2 x y plus 7 1 squared. It turns out that we can show that they represent the same integers plus m for any m, but they represent different terms. For example, 31, 31 is 27 plus 4. But it is not represented by the second one, and 37 is represented by the second one, but not by the person. So this sort of really needs to go beyond the arithmetic progressions. And now it's known that this was done very deep results of classical theory and complex multiplication. Given there is another reason for that. Given any n, we can characterize all the primes of this form. But the algorithm is a bit complicated to carry out the practice. So to answer some questions of the primes, we need to use different rules. So let's go back to the case of counter square. We define this, what's the r of n? At the number of computations of n as a sum of the squares. So a classical problem of that is to estimate the average size of this. This sum of r of n of integers up to n. So we are counting all the number of computations of n as a sum of the squares. And this is the same thing as we appear a circle. This is the radius square root of x. And we are counting all of that and count that. There's a big difference in that equation. So we want to count all the other points inside of the circle. This is going to be deltas that is approximately equal to the area of the circle. 5x. And to get another term, we go square root of x. This was later improved. But it seems to be using some idea of one line toward the area of x to the one circle. And it seems to improve a little bit more over the years with rules such as exponential sum. But still something just slightly smaller than x to the one square. So we can try to generalize those ideas. And for that, we consider positive definite quadratic forms, which in this talk, they're going to be functions of this form. So they're going to be in two variables, a, b, and c are the prime integers. And that discriminant is negative. So the discriminant is minus b. And b is a positive parameter. There is a notion of equivalent of quadratic forms. And we say that two quadratic forms are properly equivalent. If we can go from one to the other by a linear change of variables over the integers, we have determinant one. So this means that this information is invariable, so they must represent the same integers and they have the same discriminants. So that's why we consider them to be equivalent. And so the result is that the number of quadratic forms of a given discriminant must have proper equivalence with finance. And this is what we call the prep number h of management. And similarly that before, we can define this quantity rf of n with the number of representations of an integer n by the quadratic form x. Then that shows that if you compute the average size of the function given that the value, we can call it the nth problem. Because now we are counting the number of integer points inside the list. And now that will say that it's approximately the area of the list from their turn x to the one-series. This too started a little bit more over the years. For example, Lugrand and Lund in 2006, that is a quadratic for higher moments of this one. Now I'm going to talk a little bit about France represented by a quadratic form. So we define by f to be the number of frames up to x that are represented by a given form. And then I'll give what France shows that when you prove the prime numbers here and you also prove it for private and arithmetic progressions and also for primes represented by one of the forms, and it has this up and starts. It is what we would expect from a little x that it is supposed to be by this factor. And this basically means that primes that are uniformly distributed among all the people in this class are quadratic forms of a given discriminance. Okay, this to one. So now I want to study a different variation of this ellipse problem of lambda. So we want to estimate. We are interested in studying. Given an integer L, we want to estimate the average number of representations but only over integers that are multiples of L. It was first studied by Salman and suddenly G and he gave this formula. It has similar to lambda formula but that would have a function G that depends on L. It is a very explicit multiplicative function and he gave a narrator x to the one half. So our first theorem in this work is we can think of the function G as the number of solutions for this number as much as L. Our first theorem G is that we have to think of the same quantity with that we improve the error term to x to the one third. So it's similar to the work in lambda or that we could work in that. That would be a problem. But in particular we have an explicit version of it. We recovered it with the version of lambda of the solve. And something interesting here says, two persons in ten showed that the error term cannot be again smaller than x to the one fourth. Using some B method, we can use this to study prime in short intervals. So here we want to count the number of primes in the interval from x to x plus regular x. So what we obtain is we have the order of things that we expect but we obtain a bound with this explicit constant 28. So we obtain a good explicit constant here. We want to obtain a good error term above. The other problem that we want to study here is the problem of studying gaps between primes. So we want to study what the largest possible distance between the detected primes are summing the regular parts. So problem shows that the distance between primes can be at both square root of pm and dot pm. And this is the best that we can do even assuming the regular part of it even though the real answer should be much smaller. So for the past 90 years, the efforts have been concentrated in making this constant p as a smaller part. And there have been several results and the best approach so far was introduced by Canadian researchers from the Oregon using for the optimization to bring this constant to this nice fraction that is for the first time something less than one. And this was never intended to ground in arithmetic progression. So here we are interested in extending this to primes represented by quadratic forms and this is so-and-so. If we assume the general given hypothesis, then the distance between consecutive primes represented by a given form is to have the same order of numbers square root of p dot p. We have an explicit constant of 108 times the plus number. So the first step is to say a little bit about the method of the proof. So there are two main ideas in this work and the first main idea is that we have three equivalent languages. We have the language of positive definite put as a form, the language of life as we are to and the language of ideals of imaginary productive fields. And we can in some cases more convenient to work in one of the other language and we use all the languages in the work. And the second main theme is the use of Fourier analysis in this way. So first, we should have taken a formation formula that connects the objects that we want to start with another function of the Fourier transform. And that once we have this, we must choose an adequate step function that recovers the information that we are looking for in the best possible way. So now we go back to the ideas that I've been talking about in the start. Let me show you a little bit how this works like. So for this first result, we have this formula here. So this is a very good formula, but I think this is a nice formula because the important part is here on the left, we have exactly what we're looking for. We have a sum of our quantity, our x, which is what we want to understand in our purpose of here and only of the multiple of l, like we want. And we have now an arbitrary function g, like in the second c. And here on the right, we have a Fourier transform. So the important part is that the formula connects what we want with the function and the Fourier transform and starting from here, some work, we can recover our results by choosing an appropriate function. And similarly, we have this other nice formula here. This is called the hidden value. So this formula for HECA-L functions. And again, it's a good formula, but the important part is that here on the left, we have a sum over the theory of growth of the HECA-L functions. And we have an arbitrary function g. And here on the right, we've got information on primary fields. And here, this is that one of the functions of k. I'm just trying to explain here that way. So it allows us to identify primary fields. So the formula connects here with a function with primary fields of a quadratic number field and with an arbitrary function and a Fourier transform. So we select everything that we are interested in and starting from this formula with a little bit of work following the strategy of a middle and middle system that I've done. If we choose that very special appropriate function that we find on the computer, then we can recover that result. So that function that we choose looks like this. Let's take that here. And send it to you. Anyone has any personal problem? So you mentioned this result that you guys expanded this range to prove that the trajectory vary in this long range. You've got to log the portals, right? Log into the portals. So it's the main option to go beyond that. Do you expect to have some variation to your methods to work with it? So if I was in for the trajectory, something below a little bit larger and bigger than largely, right? Yes. So should I be able to go beyond that? So part of the problem is that in this conjecture that's the transaxial person, the percolation conjecture, we already have this restriction that is where it comes from in this order. So here we have below plus square root of p. And so there's an obstruction thing in the competition here to go beyond that, but we do expect it to be true in a much longer range, but it is not clear how to extend this. So if you have this trans conjecture no longer range, we're going to say yes for this range. And what's the reason that the chain did not conjecture this no longer range? Is it always the false? It's not conjecture, it's false. For chains, yeah. But yet it turns out that the family no longer worked out. It might be true in an under arrest, but I don't know. We actually emailed Chanda, asked him, so what was the reason? So he thinks it should hold in a longer range, but there's one term in his paper he has no idea how to ask to pick up, which clearly like blows up at this point. So he didn't do a comfortable saying it holds in a longer range because there's some term in his paper that obviously gets paid. Maybe it's canceled by something else. There's been, yeah. So he, I don't remember, we could go ahead and take a picture. He said in inflation, you know, 43 is the perfect pay that he's given. The difference. There were the irons. That's simple. Yeah, so I don't want the link of the name of the person. Yeah, I see what you're saying. I stop it. That's the biggest thing I've ever seen before. I don't know about this. I think it might be kind of like we're moving from that. So I don't know if it's going to worry about it. And either it's going to be very, very good. And then when we get never connected in my head, it's going to seem to be in one way or the other. I don't see anything. We have to sort out where it's going to be. Even the last one is announced. It's not going to be anything. Yeah, so if you just, if you just, you just, if you just, if you just, if you just, if you just want to keep on telling you what you're doing. And then it's going to be in two. So if you just, if you're just going to keep on telling you what you're doing. And then you're going to get it on. So I think that's the biggest thing I've ever seen. So, I think that really is going to be a little bit of a big difference between being in the company and being in the community. I'm going to be glad to say that our kids are killing themselves on average and so we're going to all be doing better.