Hello everyone, first I would like to thank the organizers for inviting me and also for having all this energy to organize these online seminars in the last few years, especially during the first year of COVID. This seminar has helped me immensely to stay in contact with the math community, so it has been super helpful and I hope it continues. Okay, so today I would like to talk about the second moment method for a rational point. So this is a method that people in analysis or analytic number theory have been using almost for, I don't know, like 20 centuries, if we can say this, but in the last two or three years people who do geometry and rational points, they use it more and more and it's almost like a small industry now, so I thought I would give some kind of introductory talk around these topics. Okay, so let me, okay, it works, right? I'm in the next page. Okay, so one important question in rational points is Hilbert's 10th problem. So this question broadly says, give me any polynomial equation with integer coefficients and say I want to solve it in the integers or in their astronauts. Is there a finite time algorithm to solve it? And this was asked, finally, by Hilbert in 1900 and Matiasiewicz in 1970 is building on the work of many, many others. Amazingly, he proved that there are some strange equations where there is no finite time algorithm and if you've seen these equations and you are an analytic number theorist, what you could think is these equations maybe are not so typical, these counter examples, I mean, they require like half a page to write and maybe they are a bit complicated. So yeah, you could ask what happens if I pick like a totally random, like a typical diophantine equation. Is there a finite time algorithm? So that's the main question I'm going to talk about today. Okay, so here is one of the very central conjectures in my area, it's by Jean-Louis Coyote-Len and it says the following. If you give me some equation which defines some kind of variety over the rascunals and if there is simple, if the geometry is single, so professionally we say smooth, projective and rationally connected, then there should be a finite time algorithm. Why? So there is the Brouwer group, this is a finite group you can calculate for every diophantine equation and just by looking at this group, whether it's trivial or not, you can decide if you believe the conjecture, you can decide whether there has a principle holds. And there has a principle also is a finite time algorithm. Why? Because if you have a smooth equation, then you only have to check so your ability at the finite list of primes. And for every prime, there is a finite time algorithm, basically Hensel's lifting. So for every prime there is a finite time algorithm to check if there is a periodic solution. Okay, so today I'm going to start by giving a few examples and then I'm going to mention very philosophically what is the second moment for rational points and in the last section I'm going to go into a bit more details about one specific infinite family of equations. So these are called Satellite surfaces and together with Tim Browning and Joni Derevine, we proved that 100% of these equations, of these Satellite surfaces, they satisfy the Hasse principle. Okay, so if you don't come from analytic number theory, one question you could ask is what exactly is a typical diophantine equation? Like how do you define this thing?  So all the coefficients go from one up to a million and then you check if every one of these equations has a solution and if it is everywhere locally soluble. So I mean, what I'm trying to say, the obvious way to define typical is to order these polynomial equations by the size of the coefficients. This is the most natural way to do that. So one example in, I guess 2014, so Manzolberg have approved that if you take a cubic equation in three variables, homogeneous, then there are 10 coefficients. Okay, so if you let these 10 coefficients to leave to vary in a big interval, so from minus a billion up to plus a billion and you let the length of the interval go to infinity, then positive percentage of these equations will satisfy the Hasse principle. And interestingly, he proved that the positive percentage will also fail the Hasse principle. And yeah, so the technology he used is a lot, he was building on a lot of the things he was doing before about averages of Selmerangs. So you know these questions even like for some cubic equations, like in three variables, these questions are not so trivial as you might think when you first see them. Okay, so one very, very recent result from a couple of years ago, so Gos and Sarnak, they proved that the Hasse principle holds for 100% of these cubic surfaces. These are cubic affine surfaces. So the equation is an integer is equal to a sum of three squares minus the product of these variables. And these are very difficult. You cannot deal with them with like, I don't know, like the circle method, because the number of variables is very, very small. And they used a lot of ideas from coin Lennstra, the circle of coin Lennstra heuristics, like averages of class numbers of quadratic fields to prove that the Hasse principle holds for 100% of coefficients k. So you let k be in a big interval, a random integer, and then you check local to global principle. Okay, so another example more related to what I'm going to talk about today is that together with Alexis Korobogatov in 2020, we proved that the Hasse principle holds for satellite surfaces with positive percentage. I'm going to define satellite surfaces a bit later. But let me just say for a moment, another result that I really like, it was published, I believe, last year in the Annals. So it's by Tim Brown in Pierre Le Bouteck and Bill Savin. So they proved that some very difficult varieties in very small number of variables satisfy the Hasse principle with probability 100%. So one of these examples are homogeneous equations in degree four and in five variables. Or in homogeneous equations in degree five and six variables. So these are like really difficult. I mean, for the circular method to work, you need at least at the very, very least the number of variables to be two times the degree. And yeah, so this result is proving a conjecture of Bjorn Pooner actually. Okay, so let me try to give a well defined question. Also, what are we trying to do here? We fix an infinite family of equations. And we want to follow the conjecture by Coyo Telen. And so what we will do is we fix an infinite family of projective rationally connected varieties with rational coefficient. And then we assume that the Brouwer group is trivial, the generic Brouwer group. Because you could just take one equation, for example, for which the Hasse principle is not satisfied, there are some counter examples. And then just take some infinite twist of this equation. But then the generic Brouwer group, it will not be trivial. So first we clear out any possible algebraic abstractions. And then we order the size of these integer coefficients of these equations, we order them by absolute value. And then the main question is, does the Hasse principle hold for typical varieties in this infinite family? Okay, that's philosophically the main question. Okay, so what is the second moment? I mean, for the Hasse principle, the first example I could find possibly there is an older example is in the 1920s, so Hardy and Lithwood, they use the GRH to show that the Goldbach conjecture holds for 100% of even numbers. So they proved that if you look at the random even number between one and X, then there is 100% probability that it is a sum of two primes. In my mind, it is a Hasse principle question. Although you can argue about this. Okay, so what are the philosophical steps for this second moment method for proving existence? Okay, so let's say you fix a subset of Zn and then you're asking, is some polynomial equation soluble in this subset? So you want to solve this polynomial equation where every xi is in this subset, maybe it's the primes or maybe it is the cubes or I don't know, it could be anything. Okay, so the first thing you have to do is you try to make a conjecture for the counting, for the number of these solutions. Okay, so you're asking how many integer vectors are solutions to this fixed polynomial equation and then typically the answer will be some function of X like X or X over log X or X square, some simple function and then you have to multiply by a product by an Euler product of some periodic densities. I mean, yeah, if the family is difficult, it's not so easy to try and predict a conjecture but sometimes you can just work on the major, let the minor arcs be zero, just equate them equal to zero and then see what the major arcs give you. This is what Hardin-Littgood always did. Okay, and then the main object in the second moment method is the following expression. So you sum over the integer polynomials f whose size of the coefficients goes up to h. Okay, h will go to infinity, it's the main parameter. Okay, and for every polynomial f you're adding a square of something, basically you're adding the square of the error term. So this is just the number of solutions minus the conjectured value from the previous step. Okay, and the substantial step in this method is to prove that on average, this thing inside the square is smaller than g of X squared. Okay, so by much smaller, I mean little low and why is this useful? Because typically for a random polynomial, this product of periodic density tends to behave like a constant on average. Okay, and then so from Chebyshev's inequality, you can show just by using the statement that the number of solutions will be lower bounded by this Euler product times g of X. And the last step, which is very easy in more analytic problems but becomes harder and harder when you go to geometric problems, the last problem is to show that if your polynomial f is, has periodic solutions, then this periodic factor Cp is positive. And actually you have to prove that this product is not close to zero as a function of X. Okay, so and if you do these three steps, then the conclusion is that 400% of polynomials f in your favorite family, if there are periodic solutions, then there are integer solutions. Okay, so let me just give you one very single example of this second moment method. Okay, so let's recall what is Schindsel's hypothesis. I'm only going to state the very old version already known by Bunyakowski. So it states that if you give me any reducible polynomial in one variable with positive leading coefficient, you assume there is no periodic abstraction. So in particular, you assume there's no fixed prime divisor. The conclusion is that it represents infinitely many primes. Equivalently, it represents at least one prime. Okay, as in many problems in analytic number theory, infinitely many is equivalent to just existence. Okay, so this is a very hard problem. It's only one case that is known when you have one polynomial of three one, and this is Dirichlet's theorem in progressions. And here, what is the second moment expression you would like to work on? Well, you average over integer polynomials of pre-fixed degree, let's say degree 10, and you're looking at only reducible polynomials of degree 10, and all the coefficients go up to eight, and then you add the error term. Now, luckily, this error term, you don't have to make a conjecture. This is the Bateman-Horn conjecture. This is the error term in the Bateman-Horn question. So we know what these periodic factors are. I have included infinity here, because c infinity is one over d. Okay, and then together with Alexis Corbogatofi in Imperial College London, we proved that on average, this error term has a logarithmic saving. When the counting parameter x goes up to a fixed power of log of h. And this is already enough if you only care about existence. So if you follow the previous steps in the previous slide, you can prove already that the central hypothesis in my mind has a principle holds for this problem for 100% of polynomials. Okay, and using ideas of Coyote-Len from the 70s, you can prove directly results from this corollary, results about the off-and-in equations. So one very, very simple example is, let's say you're looking at this equation. A sum of two squares is equal to a polynomial. This is the simplest example of a satellite equation. You want to solve it with an integer t. Okay, and directly from this corollary and using like ideas of sans-souq, certain-dier, and Coyote-Len, we proved that the has a principle holds with positive probability for this surface. The problem is there is always some loss of information when you're just using simply primes, prime problems to solve passive principle problems. I'm going to talk about this a bit later. So let me just very briefly talk about the idea of the proof. So let's say you only want to do the first moment. We shouldn't go into a lot of details. So you're averaging over all integer polynomials of fixed degree d, and then you're just adding the counting function of the number of prime values for the polynomial. So we freeze the value of the polynomial p of n. What it means is that we're summing over all integers that could assume the value of p of n. And then stupidly we then add the indicator function of the event that this integer k is equal to the value of the polynomial. Okay, and then what we do is we use the standard circle method identity to detect whether an integer is equal to zero. And if you do a few steps, a few single manipulations, you get an integral of basically the unit circle of some exponential sums. Now these are very easy exponential sums because the coefficients, they vary in an interval. So this will give you linear exponential sums. It will give you, to be precise, it will give you the Dirichlet kernel. And the primes will be separated from the polynomials. That's why you do the circle method. And what will be left of the primes will be the classical binogrado exponential sum. So you're just summing an exponential where the argument k is just a prime in a big interval. Okay, and roughly what goes on, if alpha is very close to erasional, then you just use sigil, wallfeeds, theorems to control the sum. And if alpha is not close to erasional, already just the estimates by binogrado are okay for this problem. You can upper bound this sum. Okay, and then we have this, we uploaded this paper and after a few months, there was this online meeting in Mitter-Gleffler and there were a lot of talented PhD students there. It was a good place for me to feel very old. Okay, so there was an open problem session and I asked a little problem. I asked, can we prove better error terms and better results if we don't care about the primes, if we only want to control the Moebius function. So here there's the big conjecture called the Tsaola conjecture and it says that if the polynomial Pn is not a constant times the square of a polynomial, then the Moebius evaluated that this polynomial should take random values one or minus one.  Okay, and then Jonitor Avainan, he came on very quickly with a very general method which applies not just to Moebius, not just to primes, but to any function that has some general properties. Okay, so what are the properties? Still, you have a function defined on the integers that has zero average in arithmetic progressions and in short intervals. I will explain in a few minutes what exactly that means. But let's just leave it at that for now. Then fix any number D. This will be the degree of the random polynomial. Then for 100% of degree the integer polynomials, this sum one over x times the average of f over the polynomial value, this sum converges to zero. So yeah, so in particular you could say the Tsaola conjecture holds on average for generic for typical polynomials. Now, this theorem, the assumptions and the conclusions are not well defined at all. So let me just tell you what they really mean. So the assumptions, so the actual thing in the theorem says that there exists two constants a and delta such that for every moduli going up to a power of x, so this is much stronger than single wall feet. You want the function to have zero average when you look at the progression, but even when you have small intervals. So the intervals would go from x up to x plus a small power of x, but specifically one minus delta. And okay, so x to the one minus delta is just the length q is the number of terms one over q normalized by one over q. And so you you're asking that your function f has zero average in this very strong sense and you want to get a logarithmic saving. Now you're only asking it for almost all q. And by almost all q, I'm excluding basically Ziegel zeros. So this kind of assumption is difficult to prove, but it is doable using Huxley zero density theorems and other like deep ideas. So for so it is difficult to verify this assumption, but it can be it can be done in some cases. Okay, and then what is the actual conclusion? Well, if you take x the length of the sum to be a small power of eight, namely one over two times the degree, then the second moment of these sums will have zero average. So you'll get logarithmic saving. Okay. So let me give you a few comments. Actually, this theorem, if you look at the paper, is not there because we we have the more general version where you don't have to randomize every coefficient. You can fix almost every coefficient except two coefficients, like what you would do, let's say in Barband, Davenport, Albertan theorems. So we have a more general version of this. And of course, you can ask, why do we have short intervals? Where does it come from? And the reason is, it's very simple, actually, let's say you're looking at a degree deep polynomial, a random degree deep polynomial. Let's say you write the value p of n is equal to a random coefficient cd times n to the d and so on, plus a constant coefficient c0. Okay. And now let's say you freeze, you fix the values for the input n, the integer n, and you freeze every coefficient except c0, except the constant coefficient. Okay. And now you're taking random values for c0 from one up to eight, like random in these values. Then what happens to this integer, p of n? So it moves around some fixed integer, which is of course, this fixed number. It's the part of the polynomial that has no, that doesn't have c0. And the interval around the center has size h, because of course, that's the size of c0, that's the typical size of c0. And now if x is like a power of h, then this becomes a very short interval. So the length is much smaller than the center. Okay. And then you could also ask, why do we care about arithmetic progressions? I mean, with the previous approach, with the second method approach, they naturally come up when you look at the major arts, but here they naturally come up in another way. So when you actually do this, the second moment, you will have a system of two polynomial equations. And now let's say you fix nm for two integers, two different integers, n and m. And you also fix every coefficient except the constant and c1. And now, so now everything is fixed except c0 and c1. These are like the two variables that are between minus h and h. Okay. Then these two numbers, if you take the difference, they are the same modulo n minus m. This is trivial because, I mean, yes, c0 will vanish and then you have a multiple of n minus m. And so n minus m will be the moduli of a congruence in a very, in a sum of length, basically a power of x. So you're starting to fall into the dangerous zone of Bombieri-Vinogradov. Okay. And if you actually follow the previous third method approach I had with Alexei Skorvogadov, you would only be able to do that with x to be a logarithmic power of h. So one of the improvements is that we prove this prime number conjectures in the toddler conjecture on average, but the size of the sums is much bigger. Okay. So let me be more precise. One of the theorems we have, oh, sorry, is somebody speaking. Okay. All right. So let me give the actual theorem for the Moebius function. So you fix a number, an integer d, and then you're looking at random integer coefficients, random integer polynomials of x degree d. And you're letting all d plus 1 coefficients be between minus h and plus h. Then this estimate here, which proves the toddler conjecture, it holds when x goes up to a somewhat large power of h. And as you know, maybe from other second moment problems, on DRH you cannot improve much. I mean, if you use DRH, you just increase the value of x just a little bit by a fixed constant. So you will be able to go up to h to the 1 over d. Okay. And this was the question that I had in mind when I asked Joni Thervinen about this problem. And soon after this, Tim Browning from ISD in Vienna, he joined the project because he found a way to apply this general tool about general functions with zero average. He found a way to apply this to has a principal problem. Okay. And now there is a lot of notation. I have to define what are these the often-dined equations, the satellite equations. Okay. So fix any number filled k and denote this degree k over k. Let's denote it by e. And denote the ring of indices as usual by okay. All right. And then we denote by n k over q, the node of the number filled. Okay. Now you fix any integral basis of the number of the ring of indices. It doesn't matter which, omega 1 up to omega e. If you don't like any of this, just think that k is the q adjoint i, q adjoint the square root of minus 1. And then just take omega 1 to be 1 and omega 2 to be square root of minus 1. It's the same proof and the same ideas throughout. Okay. And then the norm form is defined in the following way.  You just define it to be the norm of this element. Okay. So I mean, this generalizes the norm z 1 square plus z 2 square, which is what you get when you start from the q adjoint square at minus 1. Okay. And what people call satellite equation is the following thing. You take a norm in variables. So this is, this will be an integer polynomial like z 1 square plus z 2 square. It will have degree e. And it will have the number of variables here will be e. So the variables are z 1 up to z e. These are integer variables. And this is equal to an integer polynomial. That's what people call the satellite equation. Okay. So if k is a quadratic number field, then this is z 1 square minus a constant z 2 square is equal to a polynomial. And this is basically an infinite family of phonics. And geometry is called these, this professionally called a conic bundle surface. Okay. And there is quite a lot of work done in the 80s about the Hasse principle for these equations. The reason is the first nice counter examples for the Hasse principle were done, were made for equations of this kind. It was by Skovsky in the 70s. And his counter example was a sum of two squares is equal to a quadratic polynomial times another quadratic polynomial. There are some of these polynomials that don't satisfy the Hasse principle. And Manin in the, in his ICM talk in the 1970, he tried to give an algebraic explanation by, he invented this Brouwer-Manin abstraction. And then he never worked on it again. And the, this subject was taken up by Coyote Len and, sorry, this is a typo by Harari and Skoro Bogatov. And they proved a lot of results. Yeah. So, so this satellite equation is important because it's a, basically have a surface that you can break an infinite family of curves. And usually for these curves, you can prove the Hasse principle. So this is an effort to prove the Hasse principle basically by induction on the number of variables. That's why people in geometry, they care about conic bundles and these satellite equations. But, but still many things are basically open. Okay. So one of, of the important examples is the following. If you start with a cyclic number field extension, and if you give me an irreducible polynomial, so forget these weird counter examples of the Hasse principle, just take a reducible B, then it was proved by Coyote Len, Harari and Skoro Bogatov that there is no Brouwer-Manin abstraction for rational points. What it means is that the algebra predicts the Hasse principle should always hold for these equation. If you have a nice number field, and then it reducibly polynomial. Okay. So let's, let's talk about a few of these results. There are actually quite a lot of results. So I'm not going to focus on the irreducible polynomial equations. So the Hasse principle has been known to hold in the following cases. Actually, that's another type. I should have said the Brouwer group controls the Hasse principle in the following cases. So it's, it's when you have a quadratic norm is equal to a low-degree polynomial. So the polynomial must have a degree only up to four. And, and this was a gigantic effort using a lot of algebraic methods in, in some papers in Crele in the 80s. Actually, it's two papers and it was by Coyote Len's and Skoro Bogatov. Okay. So you can ask what happens if you have other norms coming from like cubic or quartic number fields and Salberga and Coyote Len again in the, in the 80s, they proved that if you have a cubic norm and an irreducible cubic polynomial, then the Brouwer group controls the Hasse principle. Yeah. And then if you just ask what happens if we have completely general norms, then the, the Hasse principle is controlled by the Brouwer group only if the degree of the polynomial goes up to two. So you see these questions are difficult. And you cannot have any degrees. Somehow the becomes much, much harder when the degree becomes bigger. And, and these results, it was proved in a paper by Browning and Heath Brown in, in GAFA. And then a few years later by Elza Brack methods, it was proved again by Derren Hall-Smithson way. Okay. And there's a very general result. It was proved in 94 by Coyote Len, a certain tire. This is basically one of the guiding results in this area. It assumes a heavy conjecture. It assumes central hypothesis and it shows that all satellite equations satisfy the Hasse principle if there is no Brouwer obstruction for all number fields and all polynomials. Actually, no, not just irreducible. Yeah. Okay. So, so let me just give you the last result from, from this paper with Tim Browning and Yonita Ravine. And it says the following, fix any number field and fix a positive integer D. This will be the degree of a random integer polynomial. Then you look at the random satellite equation and then basically we prove the Hasse principle for 100% of these equations. Okay. To be more precise, ordering polynomials of degree D by size of the coefficients and only looking, only focusing on, on polynomials with positive leading coefficient, the satellite equation from the previous slide satisfies the integral Hasse principle. Okay. So there are a few comments here. All the previous results I talked about are about the Hasse principle for rational points. In this paper we, we proved things about the, the integral Hasse principle. And this is a bit more recent area. What is the Brouwer group for the integral Hasse principle, for example? This is not so wrong. And Coyote Len with his collaborators has studied this. And so for example, he has shown that if you pick a completely random polynomial, then this satellite equation should have void Brouwer abstraction. If you're asking for integral Hasse principle. And actually it is a bit more subtle. So Jennifer Berg, she had a very interesting result. She proved that there are some single equations as sum of two squares is equal to some degree for polynomial. There are some simple equations like in low degree where the integral Hasse principle fails, but also the Brouwer abstraction does not explain this. So, yeah. So, so you cannot really expect to have 100% Hasse principle for all equations. And somehow restricting on positive leading coefficient, it, it kills all these problems that Jennifer Berg proved. So here are counter examples they had to do a little bit about so you believe in the real numbers. Okay. And actually I should have mentioned there are a few results. So there are papers by Mitankin which prove that if you assume that the central hypothesis, then you can prove some version of Hasse principle for integer points as well. And there are other papers. But in general, the integral Hasse principle is not so well studied, I would say. And as always, okay, you can always complain, we only look at random polynomial equations. We don't prove anything for any given equation. But the advantage is that there is nothing known for these equations for, for irreducible polynomials when the degree is like six or seven or anything bigger than that. Okay. So let me talk a bit in the remaining, I guess, 10 minutes. Let me talk a bit about the proof in the paper with Tim Browning and John Iteravine. All right. So then the idea is to try and prove a second moment estimate. Try to use the second moment method directly to the counting function for integer solutions. You should not be going through prime numbers, as I did in my previous paper with Alexis Kropogatov. Okay. So let's define the main arithmetic function. So yeah, again, if you don't like the notation, just think of this function out of n, it is counting the number of representations that you can write n as a sum of two squares. But if you have a general number field, you define it as the number of integer solutions to this equation, the norm is equal to n. And for to make things simple, we restrict the summation into a box just to avoid casps because the accounting is a bit awkward in casps. Okay. And the main sum we want to study is the following. You want to sum this r function at the value of a polynomial p of n. And your average overall integers n going up to x. So if you want to prove that p of n has an integer solution that is equal to a norm of this number field, all you want to prove is that this sum is positive. This cp of x. All right. And we will prove slightly a bit more. We will prove that it goes to infinity, polynomially fast. This is important if you want to prove stronger statements than existence of solutions. So yeah, I mean, in the paper, we actually don't just prove the Hasse principle, we prove some kind of Zarisky density result. So we prove that 100% of these satellite equations satisfy a weak form of Zarisky density. This is slightly bit more than Hasse principle and weak approximation. Okay. There is one little problem. You want to use this previous tool about functions with zero average. But I mean, these functions here, these r functions, they have positive average. So it's not obvious how to use them. Well, you have to use a model, you have to use a function, let's call it r hat, another function r hat that has the same average like r, even when you look in small intervals and in arithmetic progressions. And it turns out that there are many people in analytic number theory who have worked on this kind of ideas. So there is the Grammer Granville model. So basically what it does, maybe you've seen it, if you've studied the Grammer model for the primes, if you give me any multiplicative function, I can try to replace it by looking at what happens at the small prime powers. All right. So to be more precise, I will define r hat of n to be a function gamma that takes into account periodic solutions and some real density, some function omega that has to do with real density. Okay. So how is gamma defined? So here we do the W trick. So we will look at this polynomial equation norm is equal to a number n, modulo w, and we will count solutions. And what is w? It is the product of all primes up to something that goes to infinity pretty fast, this exponential of square root log x. And you also have to allow the exponents to go to infinity. So a bit like log log x. So if you do the primes, the exponents are always one. If you do the star, the old grammar model. But if you want to prove has a principle theorems, you have to allow the exponents to go to infinity. This is so you can cut solutions coming from maybe like modulo p square, because hence maybe you start lifting solutions from p cubed or p to the four. Yeah, so I'm not I'm not going to say anything about the omega function is it's an integral analog, a real analog of this gamma. This gamma is a local sounding function. Okay, so let's see how one can use this tool for these from the previous slides. This tool I talked about before about functions with zero other. Okay, so you can you can show that the average of our heart is equal to the average of far by using some some machinery from that is a function. I mean already for the sum of two squares, I guess there is very old work by Selberg and huli, which gives a positive level of distribution. And if you want to work for completely general number fields, there are a lot of algebraic issues. And, and luckily team, he had a paper with Brown, where they prove that has a principle for some satellite equations for all norms. So, so we can we could use some of these machinery. Okay, so if you use all of that then then what you get is that this grandville grammar model are hot minus the function aren, it will have zero average, even on progressions and in small intervals. Okay, what does it actually mean for us. So it means that this second moment will be small. So what is this moment you're summing overall degree deep polynomials whose coefficients go up to eight, and you have the square of CP minus C heart P. So what is C heart P is defined like CP effects, but you just replace R by our heart. So this is what is this sum, you're summing this fake approximation are hot at a polynomial value when n goes from one up to X. It is a bit like summing the grammar model. Or like if you're trying to start to make a bateman horn prediction using the grammar model, you would have some kind of some like that. Okay, so for our theorem, it suffices to prove a lower bound for this average of them are hard function. And so the goal for the remaining of the talk is to give an indication how you can prove that for 100% of admissible polynomials. So these are polynomials such that the equation they have an equation with periodic solutions for every prime P. So you have to prove that for 100% of these integer polynomials, the satellite equation will be such that this sum will be growing like a function of X. It's a bit like X over some powers of long. Okay, and so what is nice about this R hat function is that it has to do with things that happen locally. So things that happen modulo some w modulo some small number w. So you can study its average its average, even over polynomial values in the same way that you would use a rosette one in its sieve to work with the grammar model over polynomial values. So this step is not very difficult. You can prove that if the polynomial P is everywhere locally soluble, then this sum Cpx will be lower bounded by a nice constant times X. And this constant, it could be zero if there are no periodic solutions. How is it defined? It is defined by looking at the density of solutions of the original satellite equation modulo w. So you're looking at all vectors z1 zd and t modulo of this number w that goes to infinity with X. And you want to prove that this is bounded away from zero. So by ranking streak, you can do the following thing. You want to pound the probability that this density of solution is close to zero. So how would you do that? Well, the number of degree d polynomials modulo w such that this periodic density is say something like one over log X by ranking streak, you can upper bounded by one over a power of log X. And then for every polynomial modulo w, so you have d plus one element modulo w and you form this polynomial P of t, for every such polynomial, you're adding one over this density. Of course, this density, it could be zero. There are no periodic solutions, but you only do it for everywhere locally soluble polynomials and it's easy to prove that this density is non-zero. Okay. So now the goal is to try and bound this weird expression. I mean, in some areas, in analytic number theory, you're averaging singular series and you're averaging. It's not uncommon to see this kind of expressions, averages of periodic densities. However, in these areas, you're averaging the density itself. So you would sum sigma, not one over sigma. So this is a bit unusual. Okay. So in any case, you can prove, you can try to give some kind of cancel arguments, some kind of cancel lift argument, and you can prove that this one over sigma will be large, but only when the first solutions modulo primes come from very high exponents. And this is reasonable to expect. If there are no solutions modulo p, no solutions modulo p squared, no solutions modulo p cubed, but the first solutions you can lift come from modulo p to the 10, then this sigma will be small. And that's, and you can prove that it's if and only if. Okay. And this will mean in particular that the polynomial p of t in its derivative will have a common root modulo, a large power of the prime p. And the last tool we use is the Iguza-Zeta function. So if you have a random polynomial with d plus one coefficients, then you can look at the discriminant. Now this is a fixed polynomial, right? So if you randomize the polynomial, the discriminant is fixed. It's not random anymore, because its variables are the coefficients of the random polynomial. So you can use estimates about the illusions of polynomial congruences if you just apply them not to the random polynomial, but the discriminant of p. Okay. And this discriminant of p is a polynomial in the coefficients of p that is in general very complicated. And there's not much you can do. So you have to use some general theory which tells you that if you give me any integer polynomial in any number of variables, homogeneous or not, or irreducible or not, or whatever, then the number of solutions modulo high prime powers is not so big. And this comes basically it's very easy. It comes from the first pole of the Iguza-Zeta function. And actually in the paper we use some explicit version of this result from a paper of Lillian Pierce and Damaris Schindler, I think. Yeah. And then this means that this blue sun, it will be kind of not, it will not explode. It will have a very small value because one over sigma will be big only when the discriminant has a lot of polynomial solutions modulo high prime powers. And therefore this means that this probability, the probability of these densities to be small, this probability will be close to zero. So this proves the main theorem. So you have how that the percent has a principle for these satellite equations. So just very quickly to close, let me summarize today's talk. So the first thing I'd like to say is that the second moment method, there's a new tool you can use. This result we have with the Iguza-Zeta and Damaris Schindler team about arithmetic functions with zero average. And the second point is that to use this tool, it's complicated. You have to use it by choosing some smart model like the Grammer-Granville model, for example, or maybe some other additive model in other programs. And the last step is that the main theorem is that the has a principle for integer points holds for 100% of these satellite equations. Yeah, and that's all I have to say.


I'm alive now. Are you? Yeah. Hey everybody. J-Bear 1000 here. Hey everyone. It's Mikey 1000. I'll be right back. Okay. Okay? Yeah. All right. Oh, okay. So he's gone inside for a few minutes and we're going to start this. I'll show you the fire that we got going. Let's see if you guys can kind of see it. I'll have to move you around a little bit. There you go. Nice fire. So we went. J-Bear found a bunch of firewood online for free. So yesterday we went and we got a bunch. And then we decided to go back today. We got some more. So we got all this firewood now. So and I think we're going to go back next week sometime and the guy is going to gather some more up for us. So because we go camping, we can take it with us. And you know, we can also have it here. So we can have more fires in the backyard at night during the coolness.  So that is what we are doing. I'm back. You're back. Very pretty fire. Oh, hi Dennis. Hi Dennis. Glad you can join us. I'm talking to that. I know. You're just used to it. Sorry monkey people. It's okay. They saw my big button today and yours. Hi Kenny. Kenny's here. Kenny. The K-Man. The K-Man's here. We got Dennis. We got Kenny. Yep. Everybody's in the house. Can you guys hear me all right? Dennis says nice to be here. Awesome. Kenny says what is up in Florida fam? Uh, not really anything. Just getting firewood. So we can have more fires in the backyard. Okay, cool. Thanks Dennis. I got the mic sitting on the table in between the two of us because it sounds better than... You can always tell when someone's going live on their cell phone because it sounds like they're in a tin can. Yeah. I hate that. So I try to use the wireless mic whenever I can. I want to get one of those, one of these mics only for my camera. Instead of, you know, from that company. Oh yeah. You know. For your camera. Yeah. Thanks Kenny. He knows what I'm talking about. I know what Kenny's going to say but... Does anybody want me to do a live stream Halloween night? We can call it Shade Bear's Live Scream. Kenny says you got a nice haul of wood. Yes, we did. Yeah. Yeah. Dennis says yes. And Kenny says of course. Okay. Alright then, it's on eight o'clock. That's, you know, well Kenny knows. Kenny's three hours behind us, right? Yeah, three hours behind us. That's cool. Mark's Adventures, hey!  Good to hear from you man. Yeah. He's been watching us for a while too. Yes, he has. Yup. Alright, yeah so, yeah let's see. It'll be eight o'clock our time. Since we didn't get to go and do something.  And I'll tell you a true story or two and maybe one. I'll tell you what, I know what I'll do. What? I'll do Halloween Live Scream instead of Live Stream. Okay, the Halloween Live Scream. Yeah, he watches all our videos. Aw. Yeah thanks man, appreciate that. But we'll do, I'll do that. I'll call it the Live Scream. And I'll tell two stories. One's gonna be true. And one is gonna be totally fabricated and made up. Okay? Mm-hmm. And at the end of the video, I'm gonna let you guys give your guesses. Okay. The first person to guess it right wins a prize. Yeah? Yeah, so it's kind of like a giveaway. Okay. Yeah. And he says, yeah, it's almost like beyond belief factor or... Oh yeah, yeah, yeah! Remember that show? I was watching a couple... Fiction. Fiction, yeah. I was watching a... That was a cool show. That dark haired guy. You know there's a guy called Rob Gavigan? Yeah. He looks just like that guy. He had to change his name. It used to be Rob Dyke. Kenny says, I miss that show. Yeah, me too. But he got complaints and they shut his channel down. They just made it to where nobody could watch anything. Right. Because of the word Dyke. Really? And that was his actual name. Wow. It's on his Godgivens birth certificate, Rob Dyke. So he had to change it to his grandmother's maiden name, Rob Gavigan. Oh. Yeah. He said, yes, he had some cool dark content. Yeah. Well, he was dealing with some stuff himself, man, where he talked about...  Ooh. That's scary. Yeah. Because he was off the grid for a while. You know what I mean? He just didn't post anything. Right. Oh, hey Kenny. Do you watch... Do you watch... Yeah. Those guys? Yeah. Sure. Whenever they're called. Oh, shoot. Hang on. I'll think of it. Idol time. Hey, all right. Welcome. Evening from Boise, Idaho, my friend. When it comes to screens that doesn't get any better than the Crypt Creeper from the HBO... Crypt Creeper. Yeah. Series, toes from the Crypt. Yeah, I was watching it last night. Yeah, you watch it all the time. Yeah. I'll wake up in the middle of the night and it'll be on. Kenny said, I probably do. He probably does. Yeah. New Castle After Dark with the management. They host movies. They play old movies. Yeah. And horror movies and stuff. Yeah.  And they always got a cool backdrop and they're dressed really nice. Oh, yeah? Yeah. Cool. Yeah, you gotta check them out. But yeah, they're pretty cool. They play movies that they grew up with and stuff. And they'll do an intermission halfway through it and they'll talk a little bit. Yeah, they're pretty good guys. Kenny says he's heard of them. Interesting. I don't like all their movies, but I like them in general. Right. Sometimes I'll watch them and fast forward through the movie just to watch them guys, just to see them talking stuff. Yeah. Where's the pop at, Dennis said? Right there, of course. Heck yeah. So that's what we'll do. Yeah. That's fine. I really want to do the other thing, but we just couldn't do it. Maybe next time. Kenny says remember Creepshow. Creepshow. Creepshow. Creepshow. Yes. Yes, I remember. They have a series now and it's awesome. I watched it not too long ago on YouTube. Did you? Yeah, I don't watch many things on Netflix or anything. I'm just YouTube. That's it. I watch Netflix. I watch a lot of shows on Netflix. Well, yeah, because they have programs that you can't get on YouTube that you like to watch. Yeah. But, you know, anything they go on there just... I like real stories a lot. I like a drama. I like YouTube when it was all real. Yeah. People doing real things, you know. Was that a gunshot? No, I don't think so. It was a non-alcoholic bush. Yeah, I used to drink a lot of that non-alcoholic bush until they screwed up Anhyzer Bush. I used to drink a lot of it. No, that was a dump truck. Yeah. The tailgate of the dump truck closed. We have her gunshots here though. Yeah, he saw that video. Yeah. Yeah, because the security cam inside the house, that's where the mic is. Yeah. There's one mic for them four cameras. And it's inside the house and it picked it up. Pop, pop, pop, pop. That was right behind here though. Yeah. That was one of these guys, one of our neighbors right here. Yeah. The problem was behind us. Oh, yes. MST3K. That's right. Heck yeah. Godzilla Channel. Yep. Oh, I remember them. I remember Mr. Science Theater 3000. Yeah. Yeah. They sit in front. Yeah, I like them. And they're out in space. Yeah. I like the older ones though.  I remember when Alexis started that. Yep. Yep, I sure do. She does strange things. Oh, ain't no lie. Where's Bruno? Oh, he's in the house. He got mad at me. I sent him in the house. He got mad at you. Yeah, because he didn't want to go in. But he wanders out here and it's really dark and I can't see him. And I don't like to put him on a lead. No. Because he has to be on a lead when we're camping. But at least, you know, I make it long for him. Yeah. Yeah. But he's in the house. Because he likes, he don't just run away for no reason, but he'll chase armadillos and he won't stop until he gets one. Yeah. Yeah. If he takes off, oh my God, we can't chase him. Hey, thanks, Kenny. Yeah. Yeah. Kenny's plugging us. Thank you. Yeah. He'll chase armadillos and snakes and he won't stop until he catches them. Nope. And we're afraid one of the armadillos will run across the street and he'll be chasing it. You have to be careful on this road. They fly up here like it's a race track. Yeah. Most of those guys have probably seen the crash out there. A turk flipping. Oh yeah. Most of them, yeah, I think so. Probably. I told you about that. He stopped by and apologized, didn't he? Yeah. Yeah. Yeah, he told me. That's no problem. Don't worry about it. You know, if it happens. Oh, it wasn't my house, but it was a little bit scary. It could have been our house. So he ain't lying about that. Florida seems a little allless sometimes. Yeah. You got all kinds here. You can pretty much in this little town do whatever you want as long as you don't pull out a camera. That's pretty much it. You pull out a camera, man. And you just, well, most of them probably saw one when I had the camera out in the front yard. He was busting that guy. Yeah. Mm-hmm.  Right. You'll take my camera. Try it.  Yep. Exactly. Yeah. Yeah. He was mad too. He was like, you put that camera away. I'm like, no. I'm on my own property too. No. He said if I didn't turn the camera off, he was going to come over and take it. Yeah. He couldn't do that. He'd be trespassing. Yep. Because I had nothing to do with what was going on there. And there's no legal, legal feat how many feet you can stay away from them either. You can get right up on them and film them. Yeah. And they're way pushing them back. Right. Then that's obstruction. Right. And that's anywhere in the United States. Yeah. Well, asking Hawaii too. Yeah. Yeah. The whole Constitution. First Amendment, right? Yeah.  Yeah. Remember that one guy pulled out his card? Yeah. And on the card had the First Amendment rights on it. And he said, there's my press pass. That's the best way of handling it. Who gave you permission to film here? He said, my birth certificate gives me permission to film here. Well, Sumter County don't even have cameras. They don't even have dash cameras. No, they don't. And they told my lawyer, why don't you have them? He said, the public don't need to know what we're doing. My attorney was like, you work for the public. They have every right to know what you're doing. Yeah. Yeah. If you want the cops show up pretty quick, tell them or somebody filming you and they don't have your permission. They'll be here. Ridiculous. Yeah. I've never seen anything so different. And that's everywhere. That's crazy. They swore, the first thing they swear into is to protect, to protect our civil rights. And all they do is walk them. They don't even know what the civil rights are.  You know? He's laughing. Yeah. For instance, watch We the People on YouTube. We the People University. Yeah. He's an ex cop. Yeah, he's interested in a lot. And he quit because he knew things wasn't right. Yeah.  So the first thing they do is put their hands on you and say you're obstructing. Hey, I'm back. Sorry about that, guys. For some reason, our Wi-Fi's been pretty spotty here lately. So it's about that. It wasn't you guys. That was me buffering. I bet he's going that on purpose. Probably because I yell at him.  Go do that somewhere else. He called me a C word. Just say it. He called you a cop. Yeah. We're not little kids here. We're adults. So anyway. That's quotations. But yeah, we watch the Amagashan. I'm actually happy with him. But we watch the Amagashan press, don't we? Yeah. And another one. Jeff, he's down here on your oath. He's down here in Florida. He's another one. Yeah. He's pretty good, too.  He's a good one. Yep. All them auditors are good. And the crazy thing is, Kenny, is they're not just doing it for themselves. They're doing it for the people that are complaining to them about filming them. And they walk right up to the camera and say, I don't want to be on camera. Don't make sense. Stupid. This is the fires making him sleepy. Oh, is it? Yeah. That's what we were talking about my aunt and uncle. They live in Canada, way up in Canada. So they have to get firewood all ready for the winter. Because that's all they use is their fireplace for heat. Because the house they had is a cottage, really, on a lake. So they weatherize it so it's more of a house now.  It's horrible. I'm sweating to death and they're all asleep from the fireplace.  Because they do.  They walk right up to somebody with a camera to tell them they don't want to be on camera. But yeah, if there was a van sitting there with news and iron on the side of it, they'd be doing all they could to get in front of that camera. And they don't realize that we are the press. Anybody with a camera is the press. There's no such thing as a press pass. Right. But we're not that far from the ambulance place. No. Yeah, it was nice that guy to give us this wood. Yeah, it really was. He was a really nice guy. Do you want more of you? I'll give you more. Not just us, though.  Yeah, he's helping out his community. That's what he's doing. Whoa, like you said, I'm buffering too. No, that's not. I just said I'm buffering again. That's me. Yeah. Oh yeah, education. Yeah, we need to do it again. Yeah. Remember, we did it live. A couple years ago on Easter. Yeah. Remember Easter Sunday? Yeah. We went live. Yeah, we need to do that again. So many places to do it, too. Fun. Yeah, it's a lot of fun. It's cheap. Yeah. It's a hot one, though. Yeah, we thought about that, too. Putting it up front. What? Doing our own GK. Oh, yeah. Yeah, we talked about it. The cameras would pick it up. We should, really. That could be a playlist. People finding the geocache. Put it out there in that tree there somewhere. Yeah. Yeah. You could use that camera. One of those trees already came up. What? That camera. You could use the little camera to do that. But the trees already have cameras on them. Not on the tree, but they're pointing at the tree. Yeah, yeah, that's true. Yeah, you're right. That would be hella fun, wouldn't it? Yeah. Oh, yeah, we could do that. Mm-hmm. That'd be fun. We should. We really should do that. We should put some cool stuff in there. Yeah. Yeah. We need to get stickers. Yes, we need to do that. Leave in there. Put one in there. That'd be fun. Yeah, maybe I'll do that next week. This is coming week. Yeah, it's not hard to do, right? To put you on. Yeah, I like the... Yeah, we got the live oaks, the water oaks here, and the palm trees. Mm-hmm. Water oaks are nice here. Very nice. We do have some pine trees over here to the left of the fire. Yeah. So that's why we always got pine needles laying around. This backyard gets full of them. Yeah, I'm not so bad this year. I figure maybe they haven't fallen enough yet. Well, two of those trees die. Yeah, that's kind of... And I cut them down. Yeah. Sometimes it's raining back here and pine needles. I can't wait to get the hut done. Yeah. That'd be a lot of fun filming back in there. Yeah. Oh, yeah, thanks, Kenny. Did you get his money back? I don't know. Kenny, did you get your money back? The fire's making Dennis tired. Oh, he got his money back. Okay, that's good. Yeah. I'm glad. Yeah. I did good with that. And a lot of people is having that problem. Really? They've been having that problem for a couple of years now. Oh, yeah? Yeah. Huh? Sending. Mostly it's about sending the money. Yeah. Of what I have read that they want to help out a YouTuber or something. Yeah. And they just ain't getting it. Huh? It's not showing up or anything. Wow. But it's coming out of the banks and stuff. Hmm. Hopefully, if we're long, we can get the rest of the stuff for the truck and get it running. Yeah.  Mm-hmm. Everything rings, bearings, you know, oil pump, timing chain, gear. Right. Yeah, timing chain tensioners. I mean, that's the way to go, you know. I'm just ready to get my truck together. Because if, because if I put the head on there and get it all hooked up and started up. I know. And it doesn't, you know, it doesn't have compression in the cylinder still. That means I gotta pull the head off again. Now's the time to do it. Now's the time to put renewed rings in it, you know. Yeah, I get it. I mean, the cylinders are worn. That truck's tired. But I think a new set of rings, I can get another $100,000 out of it. You know? That's your daughter. Hey! There's my girl. Aw. Well, you tell a little man, Papo, say hi. Hi, guys. Mama monkey says hi. Hi, buddy. My grandson's watching us. Aw, he's so cute. Let me turn this around here. There you go. Hi, guys.  Yeah, Shauna. Shauna Bear. That's my daughter and little man. That's my grandson. Yeah. Keegan. Hi, Keegan. Papo's famous, buddy. I can never, there's the camera. Kenny says hi, Shebaer's daughter. See, I'm not getting mine. I'm late. You'll be honey. Yeah. Mark says it's a Toyota. You'll probably get another 200 miles, what? 200,000 miles. Yeah. 200,000 miles. Yeah, it can't. So, yeah, I mean, if we lived that long, but before, you know, she was putting a lot of money on there. That's my baby, that truck. Bear said, hey, Kenny. Kenny's laughing, too. Yeah. Kenny sent us two dollars. Yes. Thank you, Kenny. Thank you, Kenny. We did get that one. It showed right up. Yeah. Back to the park. Hey, Bear, we're going to be live. Halloween. Halloween night. Tuesday night. Yeah. So, but, so after you guys get done. Daniel. Hi, Daniel. Hi, Daniel. So, if you guys get done trick-or-treating or whatever, maybe, you know, you can join in, but back to the fire. Because see, my, I was going to hook you guys up to my actual camera, but I'm going to do that Halloween night. All right. So that way it's not brightened down, brightened down. That way it'll give me time to get it worked out. If I can remember how to do that.  I've got the hone and everything all for that. So, oh no. You got COVID.  Yeah. Hell, yeah, idle time. Yeah. Stock up on that stuff, man. I'm back guys. So sorry about that. I'm not sure what's going on. I, this first time I've had this much problem in a long time. I mean, unless I was out somewhere, but usually here, I mean, the right behind me is, is the wife. I'll show you.  Right back there, that window there. That's got the router modem and all that stuff in it for the Wifi. So I'm not sure what's going on there. I put it on 5G. Kenny's got COVID so he couldn't go to the local playhouse to watch the RRT4 show. Oh no. I'm so sorry. He's pissed. You get better. Can you hear me now, Dennis? That was us. We, we was all falling. Yeah. I made monkey. Let me see that. I made monkey a marshmallow stick. It works. Yeah. It works. Okay, cool. I even went in.  It's on Wifi in the house. Yeah. Yeah. I unplugged Alexa. The only thing that's still on Wifi is the, the camera out front. Are you on Wifi with your phone? No. You sure? Yeah. Because you should be running data because you get unlimited data. You know? I'll blur it out there for a second. Thanks, Dennis. Yeah, Daniel. Daniel's laughing. Come on. Focus. I am way out of focus now. What the? Maybe there's a ghost around here tonight. Yeah, but this is, so now my camera's not focusing. There we go. Jesus. Horrible. Something's over there. Shouldn't be. It was a full moon the other night. Over there. It's over there. Is it a full moon? I've heard it tell because the trees aren't in the way. Can you say that's better? Yeah, we'll just do what we did last time. I'll just keep it rolling. I'm at 70%. We'll just keep going until the battery gets low on my phone. Right? I'm on 5G now, which it says, you know, it's not as good. You know what I mean? Yeah. But it's acting better than the 2G, which is almost full. Yeah. Doesn't make sense. But I've had nothing but problems with this damn phone since I've got it. That's why I don't have a regular phone number. Because I ain't paying them. A $200 phone don't work nowhere near as well as my $100 phone does. The only reason why I got a new phone was because the other one, it was just so thick. It's actually bowing the screen. I thought the battery was going to explode at any time. And it hasn't done it yet. No, it hasn't. I figured it was going to be gone. Has it really bulged? Yeah, it bulged. Hey, we're in black and white now. We're in red now. Monkey said, will you make me a fork? Okay, so I made her a fork. Come on. You said, will you make me a fork for my marshmallows? Yeah, I did. I made you a fork. You made me a momsum. You gave your momsum? Oh yeah. I had to cut her off though. Make sure it tapes on there. I think the last time, didn't I use zip ties? I think so. But it wasn't a regular fork though. It was one of those cooking forks. Stabbed for stabbing meat. Yeah. No. Dennis said, now you're good. Cool, thanks Dennis. I had a time that says, last night was the hunter's moon. Yeah. Hang on. Last night was the hunter's moon. Like a damn flash light. It was cool. And it's supposed to be pretty good. The rest of the weekend as well. Yeah, I seen that we was going to have that, but we can't see it. Okay, but what he's talking about, it's not going to show up right for us. Right. Can't move the phone, huh? We'll go, it'll mess up again. Night for that stuff. And it wasn't quite the full moon. But it was close. And the clouds, there was clouds in front of it. It was really cool. I should have filmed it. Yeah. Zoom in pretty good with that camera. It's amazing. Sony AX53. That camera's amazing. It really is. Yeah, he said I should get an iPad. Nope. I like Android on anything. Apple was back in 2013, I think it was. My ex gave me one of those iPods. And I put all my music on it. Like three days later, it wouldn't let me play any of it because it wasn't bought from iTunes. It knew that it, you know, that was my music. I was like, no, that's not right. You know, so it was useless to me. You know what I mean? Yeah. Marshmallow. You think so? Whoa. That burnt out. I had a little time to say he was a big fan of the LG phone until they got caught in a big scandal and battery by the FCC. I remember that. I switched to OnePlus phones. Searched it and everybody says do this and I do it and it doesn't work. So I don't know what's going on with it. Samson don't work either. Daniel said I have a layover in Nashville on my way to Miami. Never been there, so hopefully it's not too bad. No Nashville's okay. Oh yeah. I've been there. Also can't remember if I told you I got promoted. Nobody didn't tell me. Oh yeah? Yeah. Oh wow. Got a new DM, so things are better. That's good. That's good to hear. There we go. Yeah, Dennis said he likes Android phones. Yeah, this is the Rubble 4, which is an Android and it just sucks.  Yeah, Sony. Anyway, yeah, you guys, that's all. That's the only camera I'll buy as NTU's as a Sony. I've got some, I've got those GFC's and a couple Panasonic's, the vintage ones. So that's just for my collection. But if I'm going to buy a camera to use, it's going to be Sony. Any chance I get to get a Sony, if I can afford it, it'll be a Sony. The sound system I got in the garage is, it was her dad's. It's a DVD player. It's a Phillips, I think. Is it a Phillips? Okay, don't mark me on that. I can't even remember now, but man, that thing jams. Because it, you know, it's built in surround sound. So I got surround sound in the garage. It's a pretty cool little rockin' roll. No, we sold that a couple years ago, Kenny. That was, yeah, that was her dad's. We sold it. I'm not a Ford guy and, you know, we didn't need a truck that big. So it wasn't diesel anyway. So I was just like, ah, nah, we'll just sell that truck. So we sold it. But that little F-150 had a, it had a fifth wheel on the back of it. Crazy. And an F-150, but they had a fifth wheel camper they pulled with it. It wasn't a big camper, but I got a video of it when I was looking for a water leak in it. Are those, how do you pronounce it, Xperia? Are they, are they pretty reasonable in price? Because if it's a Sony, if I can afford it, I'll get my next phone will be a Sony. Yeah. Yeah, because I think, I think you commented on a couple videos when I was working on that truck. But yeah, we sold that truck. I got a good deal on it because we sold it for 1500, which I could have got 2500 out of it. Pretty easy, but I decided just to, I said I have 1500 and her mom said, yeah, that's fine. Just get what you can get out of it. And the insurance was going to be more. Like I said, the fuel mileage wasn't going to be, I think that truck was only getting 15 miles a gallon. So I could have tuned it up and got it better than that. I don't know if you guys can hear that in the background. They're having races. We're not far from the race track. We've got a figure eight track just a couple of miles up the road. They race school buses. They have demolition derbies there. And they race where you got to pull a trailer, like a boat trailer or a camper. They race them in the figure eights. Pretty cool. There are a couple of pieces of the boat going there. We gather racing tonight. Pixel six. Google Pixel six. Not working. Yeah, it's kind of just slow. Is this okay? It's not working right there. It's not working. It's not working. Not in there. I was just helping her with her keep, she's filming too. Yeah, I'm just doing a film. She's doing film? Yeah. It depends on the model and what you want in the phone. But I've had mine long five years. Well, I'll look into that, that Sony Daniel Scott. See, he's had a little no trouble with it. Oh, yeah? Yeah, we got, well, her, the one I got her for Christmas, her little action cam is a Sony. The one she's using old blue because she messed up her good one. I dropped it. She dropped it one out of focus. I think I can take it apart and check, you know, I can get a part for 75 bucks. I think we'll do it because that's a $400 camera. And then my ax 53, which I have two of, one of them needs fixed. It's a, they're both ax 53s. Got her dad's old camera. It's a Sony, but it doesn't record anymore. It'll record, but it records black screen and you can't see nothing through the viewfinder. Yeah. So when you record, it's just recording black screens. There's something wrong with the lens, I think, in it. That's the camera? The camera camera? Your dad's. Yeah, my dad's camera. The video camera. Oh, the video camera. I want to look at that camera camera, his camera camera. I have it, but I've never really looked at it. Oh, yeah. It takes film. Yeah. It's a lot. That's your might still work. Oh yeah. Okay. Yeah. I'll look into them soon. I can't remember when you guys had that live stream when you had people call in. Yeah. I was just thinking about that too. Not five minutes ago. Oh yeah. Remember we had to call in. Yeah. I think that was a giveaway. I believe. Yeah. Kim won that thing. It was a wireless charger. Remember? Yeah. Kim won it. Remember my daughter kept calling in? Yeah, yeah. They kept calling in. Yeah. Trying to win it. And they were just so close. And then Kim called. Yeah, she got it. Because then Bambi called again, then Kim called, and then. Yeah. Yeah, I thought about doing another one of them too.  Yeah. That was funny. Yeah. We did. Yeah, that was something. I think a woman was screaming or something. Yeah. We all laughed. Well, we knew it was fake. Yeah. I mean, why would you call me? Call 911. If they don't get there fast enough, tell them there's somebody filming you without your permission, and they'll be right there. Yeah, right. That's the truth. Kim, he's laughing. Yeah. That oak's catching pretty good. Dennis is laughing. He had rust in the gas tank on the Corvette. Yeah, remember that guy? Oh, yeah. You knew. It's full of rust. Yeah. I was like, look, I don't care if it's sells or not, you know? Yeah. Well, I'm just slitting. You don't have to let me know. That's funny. Why are you telling me? You know, it's a project car. Yeah.  I don't need to sell it. And that's when I was done. I was like, hey, he's selling that no more. Yeah, I remember that, Dennis. And I was putting up Christmas lights.  We got a problem here. We don't have a problem.  And when I took that out, I showed it on camera. Yeah. When I took the pump out because the, uh, the, the fuel gauge quit working. So when I took it out, I just put, I put the whole new unit in with a new pump and everything. There wasn't a drop of rust in there because it's, it's coated. It's like having a plastic tank, you know? Yeah. Yeah. I remember that. Yeah. That was funny, Dennis. He's still, he still brings that up. And he says someone could be dying and they would, would take an hour or someone with a camera, they'd be there in a minute. One second. Yep. Yeah. That's the truth of it too. It's so weird. I don't know what their obsession of a camera is. And then the main thing, let me see your ID and they do this. They do this because he already got your ID on you. Nope. Sure don't. It's the law you have to carry an ID. No, it's not. Why are you yelling at him? Well, then he can go in the house and sit in the dark. Oh, good Lord. He's like a little kid, man. Yeah. Somebody, somebody could be on taking their last breaths. Well, we can't get anybody out there right now unless somebody's filming it. If somebody's filming him dying, we'll get somebody right there. And they always say that too. You know, it's always, they always go, you know, well, we don't have any officers available. But you start filming one officer. He calls for backup there, there in 30 seconds. I did. I thought you didn't have any units in the area. Yeah. Yeah. Or you don't show your ID. They call for backup. You got 15 squad cars there. Oh, Zeiss lenses. Yeah. That's what our cameras have in them. Yeah. Carl Zeiss lenses. That's what the X 53 have. You guys, huh? I meant to put I instead of her. See, he got you too. He's buffering. Buffering. Because he keeps saying buffing. You even wrote that. We're buffing again. We've got Karen in the house. Welcome to Shaber 1000. Do all Canadians say buffing? I think so. I don't do. I don't think so. I think some of them do. Only if they're related to you. That's not true. You're the only one I've ever heard called it buffing. It's like you saying roof different than we said. What? Up on the roof? Well, most say it like it's rough. It's rough. It's not a rough. I don't say rough. Whenever you heard me say I got to go up on the rough. No, you don't. But there are people in the south that do. Well, I'm not from the south. Neither am I. I live here, but I'm not from the south. And I'm even higher than you are.  We played in the creek. Yeah, I heard creek before. But we call it creek up there. Well, we call it a creek in Ohio. It's me that slang. It's not slang. I'll have to take you to Canada with me once. I grew up playing in the creek. How many candles do you have that you haven't burned yet? Well, I'm saving that one for Christmas. So I have that one and I put the pink one in my bedroom. But you burn them all except for? The Christmas one. Yeah. Well, they're not all burn out though. No, they're not burn out. They're all being used except for one she's saving for Christmas. Yeah. And then I'll burn it. She wanted to save the other ones too, but she just couldn't. Well, that was more of, the other one was more of a solid one. So I was like, okay, I can use this. But that other one, the gingerbread one, that's amazing smell. Matthew Perry died. He drowned. What? Yeah. Oh my God. He drowned. He's 54. That's my age. Yeah. 54. Damn. Isn't that something? I can't believe it. Wow. When I used to work CVS, I called because the police, because one of my associates came in saying there was a bunch of children's clothes in a creepy journal in the trash can. They came two hours later. So he died today? Yeah. Yeah. Friends. I love friends too. Yeah. Thank you, Daniel. I have to, I have to send you guys a homemade candle. I'm trying to, to hone my candle making skills. Hone. Hone. Hone. Hone. Hone. Oh my gosh, I need to, no. Go to school? No. I have been to school. Thank you. Okay. I wouldn't go back to that one. It was an American school.  Florida schools. Florida schools are not good.