 from Radboud University. And he will tell us about irrationality in our current times. Please, Vadim. Okay, thanks, Alina. Thanks, Philipp, and thanks, Mike, who will hopefully see me in retrospective. It's actually a pleasure to give a talk after so many days of having nothing and only the meetings one after another. And yeah, I find this time particularly irrational and it seems to be not even over, so which also means that mathematics will continue and so there will be some further irrationality since during this time. But I really hope to speak today about certain things that happened in the very last year. So at least they're slightly before the pandemic, during the pandemic and so which hopefully are still going on. And so that's kind of the main topic of my things, of my talk. I will speak mostly about the irrationality. I will also speak a little bit about linear independence and related things, different things. But I've decided to dedicate the first slide to a very special thing which happened just, which happened actually in May. So my joint book with François Brinot about the Mali measures appeared in May and it was published by the Cambridge University Press. So let me just say that to justify this page, this advertisement, the Mali measure is actually a very diaphragm topic because it was introduced, the multivariable version, it was introduced by Mali in order to prove some inequalities for the heights. They were required in Gelfon's method in Transcendence Embassy. But of course there are some things which are related to Lemme's famous question about the smallest Mali measure. And those are also related to the fountain methods, in particular to some partial resolutions of this question by Chris Smith and Edward Dabrowowski. And so therefore the topic of the book is related to the topic of the talk, but of course there are much more in the book. So I invite everyone just to see. And I even managed to include the reference to a very recent resolution of the Schindsel-Zeusenhaus problem which is related to Lemme's famous question. It was done by the Selen Dimitrov, I think it was posted before the new year. And I guess the crisis time already started at that point. So we can say, so is it okay for the, I mean for the audio? Yeah, because. Okay. Yes, Selen Dimitrov. And so I included the reference there, but of course, if you are interested in seeing this development, so you can also go to the original paper. So the book is just to prove that it's in print. So, and actually I have a poster on the book which says that there is a 20% discount. And so, but you need a special code to enter when you buy through the Cambridge University Press. And the code is very simple. So I don't write it on slides so that one really needs to see the top, to understand the code. So it's the first four letters of the title. M, V, M, M, and then 20. So that's a six letter code. So, and if you use it guarantees something like 20% discount on this title. Yeah, so that's about the Mala measures. And otherwise, I will be speaking about the irrationality of presumably irrational numbers. So in my talk and I would like to make some remarks about what is going on and to have it like in a colloquium style. So how one can prove that something is an irrational number. An efficient way to do that is actually to construct a sequence of rational approximations. But in a sense, a sequence which is good enough to prove the irrationality. So what is good enough? It means that, so you see, I mean the sequence of rational, so it's represented by the quotient of two integers. Of course, I assume that Qn is non-zero. And then I look for the sequence such that this rational approximations, Qm psi, psi is the number in question, minus Pn is first of all non-zero. Well, at least for infinitely many n, I can guarantee that. And also that this quotient tends to zero as n goes to infinity. I mean, the existence of such sequence would imply that psi is an irrational number. And this is the only proof in my talk. So I put it in parenthesis that assuming that it's actually a rational number. So we could still get this limiting relation even if we multiply the elements of this sequence by P. So it tends to zero as n goes to infinity. And in particular for sufficiently large n, so this quantity, the absolute value will be less than half, but not zero. But on the other hand, I mean, the very same quantity is an integer. So, and this contradiction means that psi cannot be a rational number. But there is more. So one can actually kind of qualitatively decide how irrational is the number to measure the irrationality of this number. If we can also arrange some kind of growth or decay of this rational approximations. For example, because it tends to zero. So, and of course this is an infinite sequence. So Qn is supposed to go to infinity. So if we can manage this inequality, so that the absolute value of this difference is less than some constants and then Qn to the power delta. And this happens at least for all sufficiently large n. So then we can also deduce from this inequality that for any other fraction P over Q, so the quality of approximation of psi by this fraction is at least of this size. Well, again, if I assume that the number Q in the denominator of this fraction is sufficiently large. So in a sense, when we can manage to construct approximations which are good, so to manage to get this delta large sufficiently large. So this would guarantee that, so that this quantity here, one plus one over delta would be sufficiently small. And so this quantity and the quality of approximations is called the irrationality exponent. So because it comes in the exponent or in the literature, you often also see this dubbed as the irrationality measure of the number psi. So in a sense, in order to prove that so psi has the irrationality exponents at most mu, so this is the notation for the exponent. So one simply needs to provide this construction of rational approximations. And so mu would be kind of, so the quantity one plus one over delta. So I don't give an explicit definition because I mean, there is no point in that. So you can, so it's some infimum of all possible things and so on. But I would mention that for any irrational number, mu is always at least two. And this is guaranteed by Dirichlet's theorem about approximation of irrational numbers. And from metric theory of numbers. So we also know that for almost every real number, two is the right exponent. So therefore, whatever you see is some strange exponents like, I don't know, five point something. Actually for all these numbers, we expect two to be the right exponent, but we cannot prove that. So any questions at this stage? Yeah, because now I'm going to do some exposition to what kind of things, what kind of objects or real numbers I'm going to consider and what is the recent developments about those things. Okay, so I should probably start with the right side of this slide because you're seeing the abstract, there is a very nice poetry. I don't know whether it's seen because on my screen it is somehow participants close. Yeah, so that's slide. Is it also the same for everyone? Then I have to read it. So before introducing, I mean, the main kind of generator of irrational numbers for today. So, and the poem is written by the author of this paper by Bruce Burns. And I think it's a real more case where some poetry appears in a mathematical paper given by those. One bright Sunday morning I went to church and there I met a man named Lurch. We both did sing in jubilation or he did show me a new equation. And then he speaks about the functional equation, the proof, which is you're not to Lurch, but to Lurch. And so there is also a story which I learned from Bruce that at some point after one of his talks, he was kind of caught by a person who told him that he had discovered a mistake in one of his papers. And the mistake was actually that in this poem, you clearly see that Bruce meant Lurch for the name which is, well, it's a Polish name, it's pronounced as Lurch. And well, that was the only mistake meant, but yeah, so I use this as an opportunity to introduce this Lurch zeta function, which is the subject of that paper Burns. But I will use it in a slightly different context because the values of this function, they appear to be very special numbers in analysis number theory. And so I give some examples like pi appear as the value of Lurch's function, also Catalan's constant, which is not known to be rational, not yet. So is also appear as the value. And there are also generalized polyloga reasons and the zeta values themselves, that's when A is one and Z is one. So we recover the values of Riemann's zeta function. So Bruce Burns had a different notation or convention. So because Z in my case is actually exponential of two pi i x. But otherwise it's the same function. So that's the main hero in the sense that I will speak a little bit about so the numbers which are involved, the Lurch zeta function. And then finally I'm going to talk a little bit more about pi. Okay, so let me start the story about the zeta functions. So we all know that the numbers, the values of Riemann's zeta function at even points or thought, sorry, even zeta values are all irrational and even incidental numbers. This is for a very simple reason that they appear as rational multiples of the corresponding powers of pi. And actually somehow astonishing mathematical community in 1978 by proving that zeta three is irrational. And since then the next breakthrough in this direction was in Riemann's work in 2000, when he showed that there are infinitely many on the list of all the zeta values which are irrational numbers. Actually he did a little bit more. So he proved that if we kind of consider the space spent over the rationales by one and all these odd zeta values, so the new dimension of the space is some absolute constant times log s. At least, of course we believe that they are all linearly independent and irrational but that was possible to do with that message. And so in this direction, there was a kind of a different method appeared recently in my joint work with Stefan Fischler, Johannes Sprank and myself. So also to attack this problem of the irrationality and we managed to show that there are actually more than these amounts of irrational numbers. And the final step, actually the latest achievement is done. It comes from China from the work of Li Lai and you who finalized their work in 2020. I think at the time where there was already pandemic in China and so they actually show that at least these amounts of odd zeta values are irrational. So which is roughly a root of square root of s. So which is of course, I mean, much more than this constant coming out from Riva's theory. So good thing is that actually they it's, I mean, this result doesn't really kind of recover the results from Riva's theorem because there is no dimension estimate. So the amount of the numbers which are listed here is really so how many irrational are, but we have no kind of dimension estimate for the space spent by all odd zetas. So it is still the same as in original Riva's theorem. And I mean, to make the story kind of concrete, so here is an abstract from the paper which was, okay, so 16th of January, 2020, it's a pretty flash result. So they actually give even some explicit value for the constant which one can put. And so this amount of numbers, C's times the root s, over log s, so I rational. So in our previous work with Fischler and Sprang, we had just a much worse lower bounds for the number of irrational. And of course, I mean, there was also work by Stefan Fischler, not on Zeta values, but on this kind of more general things where he could also come with a similar estimate. There was also a work of Sprang about the edict Zeta values. But yeah, because that was a little bit before in the story of these improvements. So I only mentioned this briefly without slide. So here is something that is not even online. So, but it was, it appeared already in several talks. And it's related to the irrationality of Zeta five. So I would recall that, I mean, apparently it proved that Zeta three is irrational. And that's the only Zeta value for which we know for sure that it's irrational, but if we go to the next one to Zeta five, so the best result, which is well way too old now is that's one of the four numbers starting from Zeta five and ending with Zeta 11. So is irrational, at least one. And so the surprise is that for the two edict version of Zeta five, so one can actually prove the irrationality. And so to, it's, of course, we cannot introduce it by the serious like I had for the last Zeta function. So because we really need something that converges pedically. And so in the two edict case, so the Zeta five can be defined as this limit. So, and of course, I mean, this is a perfect integer in France. And so the number here is, well, it's also an irrational number, so it's over Q. And so the fact that this limit as K goes to infinity exists, so it follows from classical Cuma's congruences. So this is a Zeta number. And so in the joint work of Caligari Dimitrov, so the same, the same Dimitrov, so the same, the same Dimitrov, which solved recently the Schindsen-Zeschenhaus problem and tongue. So they proved that this number is irrational. And don't think that there is a cheap way to do this in the two edict case. This is actually a very kind of significantly difficult result. And for proving it, so there was a new kind of extension of the rationality criterion introduced, which there was a school, hallonymicity criterion. So because it's not only about, I mean, the generating function of the approximations. So it's a discussion of what kind of hallonomic properties it satisfies. And depending on the order of hallonymicity of this function. So one can actually separate, so the rational and the rationality and the rationality of this generating functions. So I wouldn't go in detail because I expect that the final outcome will be published. And then also there are several talks, I mean, the authors gave on this topic. So you're invited to attend. But this is really an achievement which happens in these difficult times. So therefore it's my pleasure to actually, yeah, give it in my talk as well. The next result is about, I'm back to the Archimedean case. And this is really about this general learning function. So which I gave on one of the previous slides. So there is a history. So this is a result which would appear in one of the next issues of the Moscow Journal of Humanities among the series. And it is by Sinodavi, whom I see on my screen. I don't know whether, yeah, so. And Norike Hiratakono, whom I don't see because there are only four people on my screen. And Makota Kawashima. So in a very, actually to me, it's a very exciting development because this is a very new kind of way of constructing for the approximations to the entire set of this learning function where you allow not only kind of Z to be some fixed point and not only X to be their fixed point. So you can vary the values at which you consider Z, at which you consider the layer of the function. You can also well kind of consider different shifts, a different X in this formula and for the entire collection of the functions. So you can construct a complete of the approximation problem, solve complete approximation problem in a very explicit way. So which means that you can get all the constants very explicitly and one can show the results about the independence of the values of these functions at the points which at least Z has to be very close to the origin. This is a standard setup for this polylogarism setting. And so one can get very general results about the linear independence of the values of the large functions at several points where points are Z varying and also X. So that's what is highlighted in this paper, but I assume that there would be a more detailed account with all the details of the derivation of this result. So I highly recommend it's also on the archive, but because I have already some kind of almost final version from the journal, so I prefer to show you it's here. I hope I don't break any copyright. Okay, so then I mean kind of the last piece of information which is, which I also find kind of touchy because I know that Raphael Markovicchio was working on this criterion for many years and somehow the crisis time pushed him to actually do that and write the things down. And also to work out several examples. So what he does, so he develops a new linear independence criterion which addresses not the case of Hermit-Padeu approximation of the first kind, which is a standard setup for this linear independence criterion, but he addresses essentially mixed type or the second also the second type of Hermit-Padeu approximation. I think they're, I mean, the abstract is very long and I can give you time to read it or you can then later on follow the slides or the original paper, but well, the funny thing is that I also have kind of some exchange with Raphael about this when he was working on this criterion. And I remember that at some point I think it was in April, so he got stuck but not because there was some kind of mathematical obstacle to go. He realized that he doesn't have enough paper at home. There was no way for him to go out and there was no even way to order paper from a shop because it was not kind of considered by the government as essential thing that can be ordered online. So that was a very tough timing for him and yeah, I'm pretty happy to see that he finally put these things online. And I think that this criterion can be also kind of be crossed, Brad, with the result about the explicit Padeu approximations from the previous slide. So, and there could be some outcomes on the boundary of the two things because in the previous example, so the things is really the discussion of linear independence, so not dropping some of the things but the complete system. Well, with the linear independence criterion, we really estimate the dimension of the Q space or some over some number fields. And so this kind of ideas, they can be cross-bred and bring some new results in the area. Any questions at this point? So, I think I'm ready to go to pi and I'm not sure whether you could see again there. So what is written? Yeah, so on the right side of the slide. So it's actually a stone to commemorize from Lindemann who proved that pi is a transcendental number. And so you see, well, and it's written that in German, so he was born at, I mean, the place now is actually just grass, there is no street. And it's not far from, so who knows Hanover? So it's not far from the Leibniz Church. So I think it's about 100 meters from it. But there is no church in the case of Lindemann. It's only this stone which says that he spent his first lives in a house which was at this place. And later on he came and proved that so there is no possibility for coverage of a circle. So by showing that pi is a transcendental number. So this is in Hanover. Well, I think in Germany now the restrictions to travel are seized, I think. So one can travel, but probably it would be safer to do that at the later moments. So I don't advise traveling to Hanover right now. But the point is that, because we know that pi is a transcendental number, of course, we also know that pi is irrational. But the fact that we can write also the rationality measure or to estimate the irrationality exponent of this number. So it first only appeared in 1953, again, so the same name, Kurt Mahler. He showed that the measure of pi is most 42. Well, because pi is a so special number and because it's kind of very challenging to construct approximations to this number. Also algebraic approximations. So it was a kind of a chain of improvements which started from a work of Niyot, so who showed that the Mahler measure is estimated by 20. Then there were kind of small improvements coming from the brothers Chidnowskis. There was also one from Katerina Ruhadsar from Yornichich and Biola. And then there were several papers by Hatha, so which actually three papers where he improved again and again, so he is estimate. And finally in 1993, he published the estimate which is that the measure is at most 8.0161. And that was a very kind of serious achievement. And you can just understand. So you see, I mean, here the records were lasting, I think sometimes one year, sometimes two or three. So between Niyot and Hatha. But the next break of the record happened only in 2008, 15 years later in a work of Salehov. And so you see, I mean, the break is roughly by 0.4. And so the achievement was done because Salehov just made possible. So he found a different set of approximations. So I should mention that I mean the result of Hatha in 93. So it was a particular construction of rational approximations to pi. And so which was quite different from what was done before. So before the ideas were mostly based on this result of Mahler or under some other thing which is related to Apiree. So I will explain on the next slide. But results of Hatha in 93 and of Salehov, they were really constructing approximations, rational approximations to pi. And somehow improving on what was done by Salehov. So I'm introducing some new erismatic ideas. So with Doran-Salberger, in January of this year, we managed to break the record again. So by approximately 0.5. And to prove that the originality measure is, okay, so this is some transcendental number. It's a logarithm of some algebraic number. So the point is that it's, yeah, so it can be bounded better than it was before. So going to explain a little bit what was before 93. So because all the constructions, they didn't use rational approximations to pi, but they used approximations to the powers of pi. And so you see, I mean, if I have an estimate for the approximation of pi square, so if I have this estimate, okay, so it's, I mean, it's greater than one over U to the new. So this is valid for, say, all Q large enough. So, and this is another way to say that the measure of pi square is at most new. So that also implies that if I consider a particular family of fractions of the form P square over Q square, this is still, I mean, for PQ rational, P or Q rational, that would be also a rational number. I would get this bound as well. And then I can factor this product of two squares. And because I'm interested in the case where P over Q is an approximation to pi. So in order to estimate it non-trivial. So the other factor would be like two pi of all this size. So we would get also the estimate for the difference of pi minus P over Q. But of course it would be still U to the two new. So which means that the measure of pi can be roughly estimated by twice the measure of pi square. So if we have a good estimate for pi square, that also can imply the measure of pi. This was the trick which was used in all estimates before the paper of Hart in 1993. So they were all using some powers of pi. And pi square, so it actually, so Apiris not only proved that Zeta three is a rational but he also gave rational approximations to pi square, to Zeta two. And so the developments about Apiris approximation to Zeta two, so it culminated in a remarkable paper by George Han and Carlo Viola in 1996 where they managed to prove this estimate 5.44. So where they introduced a new arithmetic method they called the group structure arithmetic method. And with this method they also got, I mean the record measure for Zeta three for the rationality measure of Zeta three, which you see is very, very close. And the reason is that the group structure is much richer for Zeta three than for Zeta two. And I can only mention that actually, so for Zeta two, I mean this record is also now broken. So it's using a completely different construction of approximation to Zeta two, not coming from Apiris approximations. So it's slightly broken, but for Zeta three, I mean this measure of Han and Viola is still best possible. And last year and published actually this year, so there is also a measure for pi to the four which uses a version of this Han and Viola's method. So, and this is my joint work with Marco Vickio. So we proved that the measure of Zeta four or of pi four, so is at most this size. Of course for pi, it would imply that the measure of pi is four times this constant, which is not interesting. And yeah, but also for pi square, this measure is not very exciting because it would be twice, five points. So it would be much worse compared to what was obtained by Hat in 93, but at least I mean to mention that there is some development about powers of pi and it's completely different because the approximations required are completely different in those cases. Yeah, so I give this example. Okay, so a few words about what's did Hat and so what's how this construction goes in order to compare it with what was done later. You see, you take this particular function and A1 and A2 are the numbers. So it's two and one plus I. So where this is just a imaginary unit. So the square is minus one. You consider the two integrals and actually these two, they provide simultaneous very good rational approximations to the numbers log two, but also to this number, which is a complex number. So it has a real and complex part. But so if I would like to be more explicit, so the point is that, so if I multiply by a certain number, so and their multiplication is done to manage all these QNs and PNs to be integers. So in order that I can, I really have approximations with integer coefficients. So then QM in both forms is the same. So even, I mean, I integrate two different points. The QN is the same, which is corresponds to the simultaneous approximations. The PNs of course will be different because I approximate different numbers, but QNs are also just integers. So they are not, they don't have I, so they are pure integers. While PN and PN dash, so this kind of extras, not integers, but Gaussian integers. And so to explain what the kind of this multiple is, so it comes from the least common multiple of the numbers from one to three M in this case. And so we can take off from this least common multiple, all the primes. So essentially these products are such that the fractional part of M over P, so if you don't see on the screen, this is the fractional part of M over P. So lies between one over two and two over three. So I mean, this estimates for the asymptotic behavior of the D3M, so it's from the prime number theory, but also the prime number theory can be used to estimate what is the contribution of this erismatic gain by M. So it's asymptotically, so it's less considerably less than three, but still it gives a huge saving. And so that were approximations of HATSA. And now, I mean, I write the integral which was used by Salihab. So now it's really involved. So you can see now that there are all kinds of these Gaussian integers running around in this integral. And actually when I write plus minus plus minus, I simply mean that all four multiples for all possible choices appear in the integrate. So it means that on the top, I would have five factors and everything is raised to the power three M and in the bottom, so I have only two, so which are raised in five M plus one, both. And so what Salihab showed, so it's absolutely astonishing to believe that this integral is a linear approximate, sorry, a rational approximation to pi. So he actually showed that if you compute this integral and multiply it again by something, some suitable factor, so I give it explicitly, for even M, so he only used the even M, so you would get approximations to pi. So what is this factor? So okay, so this is two in the negative power, so we can save some negative powers of two. So by the way, M is even, so therefore there is no problem in this division. So then we will also have to kill some denominators coming from five and then D five M is just the least common multiple of the numbers from one to five M. And so this tilde five M is very similar to the one in Hatter's case, but the fractional parts of M over P are now between one over three and two over five. And with these approximations, he managed to improve this result of Hatter. By the way, in Hatter's case, the approximations were not only to pi, but also to log two, so because of the simultaneous ones. In this case, they are not. So they only approximate pi itself. And if you ask me what happens for odd M, so I can say you that there would be a different chain of approximations, not to pi, but to the extension on one over seven, a completely different number. Okay, so what we did with Doran, we actually, before investing so the time and trying to kind of gain some intuition what are the best things to try. So what we did, we just put different exponents, capital A's and capital B's. So we use all the symmetries of the original Salihab's construction. And then for different A's and B's, we try to see what kind of approximations we get. So it would be again approximations to pi, at least if we go along even integers M. And in order to construct them explicitly, so you say if you try to fit this expression in MAPL and to compute this integral for large A and B and M, so they wouldn't be able to do that. But there is a nice way to actually deduce some recursions, polynomial recursions for these expressions. And with these recursions, we can easily compute so their approximations and then make decisions of what are the denominators, what is the growth of these numbers. And then we can empirically estimate what kind of measure for pi we will just get by looking what deltas come out from large enough M. So large enough, so it's roughly, we would write M up to 600. And okay, so we were considering only even M in the original run. So to determine the recursions, I mean, there is an algorithm which goes back to 1990, which was in a joint paper of Anquis and Sahlberger. Now, I mean, it's even more developed, but the point is that it runs fast, gives you a recursion. And then in a couple of seconds, you get all the terms up to 600. And then you can check whatever things you would like to check about these approximations. And with this running, so we realized, I mean, so the A and B, so this hard is exponent A and B. And in Salihaf's case, A was three and B was five. So we realized that, so with the choice A equals two and B equals three, we have much, much better seen. So we have the delta, which is larger. So then the one we could get from Salihaf's case, A equals three and B equals four. And yeah, so that was the winning family, A equals two, B equals three. But we also had some other places, like five and eight was the next one. Then number five was eight and 13. And you can suspect that the Fibonacci numbers comes into play, but actually, so number six on the list was seven, 10. And so it's far worse compared to the previous examples, but yeah, so at least. So that's kind of the things that one could expect if we prove the things figures. In this case, A equals two and B equals three. A good thing was that we didn't need to look only for even integers n. So we could also use it for m old. So because in that case, we would also get the approximations to pi. And so here is a little bit kind of idea of why do we get so pi in all these kind of integrals? And of course, I explain it on this winning family. So what I do, I actually, I shift the variable by five. So in order to make the integrant more symmetric. And so this is the integrant, so which is required to be integrated. So five terms, the products, two terms. But of course, if I converts them, so if I multiply by this all using all these conjugations, I will use a completely real expression. And so this is a rational function with the numerator having degree larger than denominator. So we can expand it into partial fractions. And so that would be a form of this expansion. And because there is a symmetry of this, so that their A j's would be the same for plus and for minus sign, but also the polynomial P here would be just the polynomial in X square. So not in X, but actually in X square. And you can see it from this expression that everything would be like having integer coefficients. I think A j's, okay. So they will be rational, but P x would be from this expression having integer coefficients. Now we need to integrate individual pieces. And if I integrate there, I mean, the polynomial and just between the two values, of course I will get, so because powers of X, if I integrate, so it would be another power of X. So then when I evaluate, there will be some rational from this portion built. And the same would happen for these expressions. If J is non-zero, because I mean, every antiderivative of this fraction would actually result in some other power of five plus X, or five minus X. So I think this is also for the minus sign as well. But if I do this for the zero's term when J is zero, so I need to integrate this expression, then magically, oh, oh, so it's not seen what happens after A naught. So this is actually pi i over two. So the result of this computation of the logos, it gives you pi times i over two. So that's the magic where the pi comes from. And because we also have i in front, so that there would be only pi over two left. And so with some clever kind of arithmetic analysis, and so with some new ideas. So because in Salihab's case, he couldn't manage to pull the, I mean, for this choice of the parameters, couldn't get the right arithmetic of the coefficients A j's, which appear in this partial fraction decomposition. So we managed to prove that, well, if we multiply by this pre-factor, so we have d four n, so the least common multiple of the numbers up to four n. We have a negative power of two, so which can be saved. And we also can save by this standard factor phi n. And surprisingly, phi n is essentially the same as in Hatter's consideration, because this is also a product of the primes, which have between half and two thirds. So it's much kind of simpler, and it links it back to Hatter's case. And so to just kind of give you a short of what exactly was the novelty, so the arithmetic one. Well, of course that's a little bit longer, but I don't want to bother by technical detail. So the point is that we managed to cast the coefficients A j differently. And this cast is only possible for this particular choice of the parameters. And so what we did, so we could write A j as some linear combination with these coefficients of these binomial expressions, the binomial sum, which is involved here. So it was using some hypergeometer transformation can be written in a very different way. And this expression, if you forget about this minus j, and so j is the same as in I j, if you forget about this. So then, I mean, this sum is actually can be recognized as what is called the super Catalan numbers. And the point was that so in general, so these numbers are kind of close by their properties, but we cannot write a simple expression for them as the quotient of binomial coefficients. But what we managed to do, so that's if P is from this special set, which we gain from, and P is congruent, I mean, G is divisible by P, then so we could show that the P-addict value of this sum and of this one, so it's the same. And for that, so we use kind of classical termos theorem. So one minus T to the P is one minus T to the P, model of P. And yeah, so I mean, the technicalities are quite involved, but the point is that we really use some kind of hypergeometric and combinatorial ingredients, and they are only available for this set of the parameters, this particular one. Okay, so making the story short, I can also say that with this machinery, we can also try, well, we could do, so the computation for some other constants. For example, we can do for log three. So by considering this integral, they are of the same shape as Saliha's integral for pi, but with our choice of the parameters. And so we could manage, so this is on the bottom of the slide with this construction to prove that the measure of log three is 5.71, blah, blah, but actually Saliha have already proved a better result choosing a different set of exponents. And Saliha have managed to do that before pi in 2007. Later he was scooped a little bit, so you see there was a slightly better result coming from improvement of his integral. But finally, two years ago, he also came out with a further improvement, thinking the fourth place after the decimal. And so this was a joint work with his former PG students, von der Ilen Lutsche. So therefore this result that comes out from our construction is not as good as in the records for log three. And on the final two slides, I would like to show a table, and this I downloaded this morning from the website, Mass World Wolfram. So there is an article on the rationality measure and there is a table of that some constants are given and some best bounds are given and then so the references when and by whom they were done. So well, I mean, the last two on the list are actually the Q logarithm of two and the Q harmonic series I didn't discuss in my talk. So, okay, pi, pi square, log two, log three. So actually, I think not much is correct in this table actually, even in the first entry, I would say that it's 2020, that would be the right. And probably the last name is, yeah, it's a little bit misspelled. And also there were some corrections for the other records. So therefore I give you an overview of the records which I believe is correct. So for the pi, so this is this here and this is the work with Doran. For pi square, it's actually, it was improved. So I mentioned that in the talk, log two, it is indeed from Markovic in 2009. Log three is better, Salihov with the company. Zeta three, I mean, best possible is still, I mean, you, Touhan and Viola. And then also for log Q, so it was mistakenly given the smaller one, which is, okay, this is kind of typo in that table. And for the Q harmonic series, there was also an improvement. So here is the state of art and here is the table corrected. And by this, I just thank everyone for joining this talk. Thank you so much, Vadim for your one.