 So today I'm going to talk about moments of class group distributions. So we were interested, last time we were talking about averages of functions of the P-CLO subgroup of class groups of, say for example, imaginary quadratic fields. And we also talked about the real quadratic case, so that would be the same thing here. So we're taking imaginary quadratic fields and we're saying over however many there are, if we average some function of the CLO P-subgroups of their class group, what does that look like at an odd prime P? And I'm going to write this with this sort of curly E-like expected value or average of this function of the class groups of P to the infinity. And the reason I made the curly E and not just the normal blackboard bold E for expected value is because this is a little funky because of this limit. It's not actually an expected value over some distribution. It's only the limit of expected values over these distributions. And of course for this to make sense you have to know that I'm taking, what case I'm taking and how I'm ordering them. But in any case. So this would be this sort of empirical average over class group distributions. And so far last time we were talking a lot about what proportion of the time which you get a certain CLO P-subgroup of your class group. So that would be taking the average of a characteristic function of a finite abelian P-group. The function that was one when your group is that group and zero otherwise. And if we simply had a distribution, a measure on finite abelian P-groups, then knowing the averages of the characteristic functions would determine the averages of other functions. Because there is this limit here, there are some funky analytic issues and even in fact if we knew the averages of the characteristics functions, it doesn't determine the averages of other functions because of this limit. And there are some exercises coming up with counter examples to this sort of thing in the notes. So that's like one, you know, you might think well do you really need to average other functions besides just the characteristic functions once you know what proportion of time you get each group, then you should know the anything you want but that's not quite true because of this limit. And so we're going to talk today about another very important class of functions to average over CLP subgroups of class groups and these are so for a fixed group B the function asks how many surjective group homomorphisms do you have to be? So these are surjective homomorphisms and the function F is an integer value function that just counts those surjections. And these averages are important enough that we are going to call such an average the B moment of the distribution of groups. So you can talk about that for a, you know, a literal probability measure on the space of finding the BMP groups or also in this class group case where we take a limit of distributions, we'll still talk about its B moment and this, these moments are in analogy to the classical moments of distributions of real numbers or of random variables valued in the reals and in that case the moments are indexed by natural numbers k and the k moment is the average of x to the k where x is from the distribution or the expected value of x to the k if x is a random real number and so these in the probability theory of real numbers these moments play a very important role and there are two big reasons for that. One is that they're often more accessible in computation essentially because of the linearity of expectation or of averaging often one can get a hold of moments of distributions and secondly because it turns out that if you know the moments of a distribution of real numbers and those moments don't go too fast they actually uniquely determine the distribution that gave those averages and so there's a little bit more about that in the notes and in this case now we're not talking about distributions of numbers but distributions of finite abelian p groups and these averages of the number of subjective homomorphisms to a fixed group play a similar role so here is an example a theorem so I said this in the language of random groups so let's say x and y be random finite abelian groups so by that I mean random variables valued in the set of groups not like in elements of a group but in the in the set of groups so if for every finite abelian group B we have that the expected so this is the average the expected value number of subjective group homomorphisms from x to B is the same as from y to B and if you're not as familiar with the language of of random variables and you are just thinking about a distribution of groups this average would be you know the integral of x you know it would still just be this average where x comes from from your distribution so if you were thinking about a measure of groups this would be what this average is I've just said it in the probabilistic language all right so if these averages are the same for every B and those averages don't grow too fast so say not more than a constant times the size of the second wedge power of B so that just is the normal wedge product as Z modules if that's the case if x and y have the same moments and those moments don't go too quickly then for every finite abelian group the probability that x is a is the same as probably as y is a so x and y have the same distribution so this shows that these averages have this this critical property of moments of distributions on the real numbers which is in fact they sort of contain all the information of the distribution at first it might just be like well they obviously have some information their averages over it but you might you know you might not have first guess that they in fact have all of the all of the information of the distribution now as I said before we have this kind of limit as x goes to infinity and so we're interested in limits of random variables valued in groups or distributions and so we need to maybe be we'd like to be able to recover a limiting distribution from limiting moments and in fact that's one of the the reasons that we well for simplicity stick to the C LOP subgroups so an analog I should say of this theorem for finite abelian groups is not true if you sort of introduce the additional layer of limits as we're about to and there's some exercises in the notes around around that but if you restrict to abelian p groups so we have p a prime and let's say we have one random abelian p group and then we have a sequence of of random abelian p groups x1 x2 now this says if for every abelian p group b the limit of the the limiting moments in my sequence are equal to the moment of some fixed y and they don't grow too fast then for for every finite abelian p group the limiting distribution the limiting probability that you get some group a is the same as as the probability that y had a so this is is like the last theorem but it lets you work through a limit which is important in this context because we when we're look talking about these distributions of class groups of number fields they're always limits of distributions that not you know an honest probability measure on the on the space of groups so all right so all right so the moral of that is just that the the averages of these these number of subjective homomorphisms to be for every b actually they actually determine the averages of the characteristic characteristic functions all right so that was sort of the second property that I said that moments from the probability theory of random real numbers had that was useful was that they actually determine the distribution and then the first the first thing I said was that they're useful because they're they're often accessible they're more accessible and it turns out that that is that that is true here as well and one of the main reasons is that the moments are actually very closely related to field counting so the question that we were talking about in the first lecture all right so I want to explain the relationship of these moments to field counting so we'll start with a number field K and we let H be the Hilbert class field of K which we were calling before also K on add the maximal unramified abelian extension of K and the reason we're talking about that is because that is a particular field that we know it's Gawa group over K is the class group of K so oh I wrote K I wrote that it was the I'm gonna do the quadratic case here I should I will change this to this should be degree 2 we'll take a quadratic number field okay so we're gonna take a quadratic number field and I just know that hanging around somewhere it's got its Hilbert class field whose Gawa group is the class group of K and now what these moments they cared about surjections from say the class group of K to B all right well what what you know once we have this this particular field whose Gawa group is the class group of K via Gawa theory such a surjection exactly produces for us so Phi exactly gives us some field here L where L is inside the Hilbert class field and L over K is a B extension right so this this via Gawa theory I mean we you had to use class I mean we sort of use class field theory to produce this this extension here but after that the this surjection just via Gawa theory gives us some field L all right and related to the computation we talked about in the case of genus theory class field theory it will tell us that L over Q is actually Gawa I mean that doesn't have to be the case when you say take a quadratic extension and so L over K is a is a Gawa B extension and but not every not every B extension of K will be Gawa over Q but class field theory because it's coming from this from this age over K does tell us that L over Q is Gawa and that the Gawa group is a particular group it is the semi-direct product of B with the Z mod 2 Z where the non-trivial element of Z mod 2 Z acts on B by multiplication by minus one so that's this minus one down here I'm telling you how the Z mod 2 Z acts so what happened we went from one of these surjections and we got we got out of it a field a field that was Gawa over Q with a particular Gawa group and conversely if I have an L over Q that's Gawa where the Gawa group I'll sort of make the Gawa group diagram over here is the semi-direct product of B and Z mod 2 Z you know because of you know the the the subgroups of this group you know inside of here I can take a you know an index index to to subgroup B and that will give me some that will give me some quadratic field so since this is an index to subgroup I get a quadratic field and since this is B by you know Gawa theory so this is this is just you know L fixed by B and then this is a B extension so this this was starting all right it's like the same you know like why is she drawing the same picture the point was this picture just started with a Gawa L over Q isomorphic to to B semi-direct product Z mod 2 Z extension and then it produced a quadratic extension and a B extension on top of that and now we had one other thing that our our L's that that came up here okay so the the this is the maximal unramified abelian extension so B is of course an abelian group here from out of a suggestion from the class group to it so the abelian part is taken care of but we also know that this that this L over K that we get here will be an unramified extension because it's sitting inside of the Hilbert class field there and so we can just see from the Gawa theory and the theory of the inertia group that L over K is unramified exactly if all the inertia in the Gawa group of L over Q intersects trivially with B all right because we want the B extension sitting there we want the B extension to be unramified so there should be no inertia no non-trivial inertia subgroups there okay so then that tells us that the average of the number of surjections from class group of K to B is actually the number these surjections correspond to these L's so it's the number of L over Q the G extensions for this group G such that dot dot dot and the dot dot dot here is basically just basically just this condition here because we wanted if we want the L over K to be unramified we need to put in this condition so that L over K is unramified now also if your average meant to be over imaginary quadratic fields the dot dot dot would also want to include the condition on the decomposition group at infinity so that you got that quadratic field is imaginary but so that you know that's that's the numerator that this average number of surjections and then the denominator is just the average number or sorry just the you know the number of quadratic fields and all of this is sort of up to discriminant x and limited x goes to infinity but basically this turns the numerator and the denominator of this average both into field counting questions now we already knew that the denominator was a field counting question and so the numerator is also a field counting question with a slight twist that we're going to maybe impose these conditions on inertia groups and um and decomposition groups now we haven't discussed that um very much except that we did we you know we have already we split our quadratic fields into imaginary quadratic fields and real quadratic fields so that's imposing some condition at infinity but a natural follow-up to these uh to these counting questions for county g extensions are to also count g extensions with various local conditions like oh the inertia should be here decomposition groups should be here um and i've given some references in the notes to to much more discussion about the variance of the counting questions where you're counting with local conditions uh but yeah so in any case this turns the question of finding the b moment into just two count two field counting questions um one in the numerator with some some additional conditions on on inertia groups and then one traditional one as we already had in the denominator and uh so this is not only a nice relationship between the class group questions and uh the field counting questions but also the actual only average that we know um i keep forgetting to put my infinities in here the only average that we that we know i mean i suppose you could average like the identity like the all ones function but only non-trivial average that we know on silo p subgroups fraud p of class groups of quadratic fields precisely uses uh this this relationship so um the only and uh the only thing we've been been able to access at all uh it via proof uh in this setting uh comes comes via this relationship between the moments and and field counting and so what is that so um that's the theorem of downpour and how brawn uh that says in this case this case that we were considering um today uh was that over k imaginary quadratic which says that this average remember this is like the limit as x goes to infinity of the average the average number of subjective um group homomorphisms from the classical of k to z mod 3z is 1 over k imaginary quadratic fields they also do it for k real quadratic fields and then you get one third but we're we'll just um focus in the in the lecture on the imaginary quadratic case and again in the notes the all the also all the details about the real quadratic case and so perhaps you recall i talked about um uh david we're a heilbron in my first lecture or very very briefly gave you a sketch of how they approached counting cubic fields and so in this case these these uh l that you need to count for the numerator just say get this z mod 3z moment right we have this similar product of z mod 3z with z mod 2z by minus one that's just the group s3 so they that so one just needs to count s3 gawa extensions but of course s3 gawa extensions there are sactic extensions but they correspond um to non gawa cubics they all have like one isomorphism type of non gawa cubic in them and every non gawa cubic you know has a gawa closure um and so uh we already talked about how they could count non gawa cubics and so they can count s3 gawa extensions which is almost what you need for the numerator here except um that they also were able to do this imposing conditions local conditions on inertia um which is what you need to to get this sort of you know maybe i should say inertia conditions maybe there's also a condition at infinity um here all right and so that counting uh so it turns out that you know they were remember they were counting lattice points in a fundamental domain and uh these inertia conditions are are mod p squared conditions on the coordinates of the lattice points for every p and so you know they just have these um sort of uh sub sub uh lat and well sub lattices or transites of sub lattices uh that that they have to count and there are some issues there are some some things that one has to do with the getting that all to work out for infinitely many p and the error bounds um but they do that and then they were able to prove um that this average number of subjections is one all right and so this uh is indeed as predicted by the Cohen-Linster heuristics so we um uh yeah so as you would hope especially because this theorem of Davenport and Hybron I mean I failed to put years in everywhere but this predates the Cohen-Linster heuristics so certainly the fact that this was already known uh was in in one case uh was a was a helpful helpful piece of evidence um and you know shortly uh of course uh just by uh sort of kind of meta thinking of course the Cohen-Linster heuristics must have predicted this uh since they wouldn't have put out conjectures contradicted like the one theorem which still remains the the one theorem all right but let's see um yeah so what does that mean that it was predicted by the Cohen-Linster heuristics if you remember um how I described that previously I said oh well the the empirical average over class groups uh is predicted to be that just this group theoretical average over a billion p groups of where every billion p group is weighted by one over its number of automorphisms so when I say that this theorem appears saying that this average or moment is one I was predicted by the Cohen-Linster heuristics I mean um uh that that that the average the group theoretic average is one for every um a billion p group b so that's a kind of cool thing also about the uh Cohen-Linster distribution for uh imaginary quadratic fields it's the distribution you know this is the distribution on a billion p groups whose whose all of whose moments are one um and one could actually um work that work that out in a number of different ways um and Cohen-Linster's original paper they they develop some great machinery with with generating series that can help average you know average all kinds of functions uh with this weight because of course once you make a conjecture like this you'd like to actually be able to to see what it it says in various cases so that's a very powerful machinery um that you could use to for example to to find to find this this average um now but I since we talked before about this matrix model I'll just give you maybe a quick way to think about uh to think about these averages being one so remember last time we took a hard random matrix uh from uh in by in matrices over the periodic numbers so let's think about that uh that matrix and then we took its co-kernel uh and and thought about that as a random a billion p group and so what what are the moments of of those co-kernels so if we want the the b-th moment the expected number of projections from the co-kernel to b so what is that co-kernel it's just zp to the n mod the image of zp to the n under n or otherwise this is like this is the you know this is the column space column space of n all right um and this is actually you know last time we very briefly talked about something that was where we were essentially uh computing the expected number of isomorphisms from this to b and so actually computing surjections is like just what we just stopped one step short uh of that that argument um so so every such surjection would come from a map from zp to the n to b so we can sum over those maps and we can say what's the probability that this column space is actually in the kernel um and as we discussed yesterday when you take har random uh piatic matrix in biometric matrix each column is independent from har measure on zp to the n and so the chances that each column lands in the kernel is one over the size of b and you have n of those independently so you get this this sum over the surjections of b to the minus n and so how many surjections are there from zp to the n to b you can actually write that down explicitly but especially when it is large it's practically the order of b to the n of them because you know most of the time when n is very large compared to b as you throw down um uh in elements of b there'll be enough to generate b so these moments of these co kernels of the har random matrices as n goes to infinity they go to one um so so the limit of the moments uh is one and now it this doesn't automatically give that the moment of the limit distribution is one um and why would we want that because we said last time the limiting distribution is the colon lindstra distribution um because to to to get between what we set up here and this statement you have to move the limit from the outside to the inside of this expected value uh which is an infinite sum and you can't just arbitrarily move limits uh however morally it's perhaps the i think the the most convenient way of thinking about uh the fact that you get that these moments one because you can really sort of see it in these roughly b to the n maps from zp to the n to b and then each of them has a order of b to the n uh chance of surviving and actually being a map from the co kernel um and indeed one can use this and this is uh done in the notes with a convergence theorem like dominated convergence theorem uh you you can interchange the limit and the expected value like i said one doesn't just have that this is a limit but you can write down this number of surjections very explicitly uh and then you can use that um with a convergence theorem to show that indeed uh the moments of this colon lindstra distribution are one any questions okay so um all right so just in summary if uh we knew overclass groups that the average number of surjections and man i really don't like writing these infinities in here from the co p subgroup of uh our class groups uh was one for every b then we would uh uh we would actually know this is from this moment moment determining the distribution theorem that i mentioned before we would actually know that the proportion with any particular co p subgroup of their class group was as predicted um but as i pointed out at the beginning but not vice versa uh so so and because of because of the appearance of the limit in the whole whole story uh so in that sense knowing these limiting moments is it's it's strictly stronger than than knowing the limiting distribution it implies uh the limiting distribution all right so in the next um lecture is it my final lecture tomorrow i'm going to talk about the generalization uh of these conjectures to class groups of higher degree extensions so so far it's been all class groups of quadratic extensions just because that was a good place to start um but we're interested in of course class groups of higher degree extensions as well now um um because of the fact uh that um uh the maybe i should say here a class group sort of class group elements elements of class groups of quadratic fields correspond to orbits of binary quadratic forms so i'll just say so this is a this is a correspondence as far as i understand that that's due to denokin so you often hear people talking about oh gauss conjectured this and that and this about class groups of quadratic fields gauss was really conjecturing uh about uh the orbits of of binary quadratic forms and now we just translate that into the language of class groups but because of this um because of this relationship and then us having a good reduction theory over here on the binary quadratic forms uh there are very large tables of class groups of quadratic fields so a lot of data available um and that uh you know data has played an important role as i mentioned before in both sort of uh inspiring you know creating a need for conjectures to sort of explain phenomena that were being observed empirically and also to provide evidence uh for those conjectures and these very large tables uh show that this cone lister conjectures for quadratic fields look really good i think there's every every reason to to believe them um exactly exactly as i've stated and um so in higher degree uh one does not have this same this same correspondence um and so the tables are much smaller uh at computations of are much slower and all of the kinds of statements that we've made so far about distributions of class groups of quadratic fields and the statements that we're going to make um tomorrow about higher degree fields are all just limiting statements their statements about limits but they don't include any any conjectural information on how quickly that limit is going to be approached or any kind of speed of convergence we say the limit of as x goes to infinity is supposed to say be one but we don't say how quickly were we planned to get there um and this introduces significant challenges uh in using empirical evidence for conjectures of class groups distributions of um of higher degree fields and and that is actually really important because unlike this situation where these uh these quadratic field conjectures look quite good there are there are various anomalies and issues with the higher uh degree conjectures some of which we will discuss tomorrow and so you would really like to so there's a much bigger space for for good empirical evidence uh to to play a role in understanding those conjectures so why am i talking about this um today uh uh before i i talk about all these other conjectures because i think that there's something important to be done in the quadratic case and um and that that is i think that in the case of the colon linscher heuristics for distributions of class groups of quadratic fields where we're pretty confident that the limits are are approaching the numbers that we we think they are it would be it would be excellent to have good heuristics predictions conjectures for things like the speed of convergence or error terms or secondary terms um um you know for these colon linscher conjectures here here i'm talking about four quadratic fields as a start um because this is a case where we're pretty confident in the in the actual limits and that makes it much easier to to be on solid ground to try to develop a theory or even just predictions from computations from tables of uh how quickly we're going to get there and then you know especially how uh does this sort of speed of convergence or error terms or secondary terms depend on which moment we're computing or maybe which group we're you know averaging how often it happens um and i think that this is especially interesting if it can be done in a way such that it would give insight into what we might imagine for higher degree fields um and so maybe it's a little bit uh you know confusing if all the real questions about what is is going on or not are for higher degree fields why would you start here um uh where the where where we believe very solidly in the conjectures it's because um it it's it's very hard to uh make guesses about the speed of convergence when you're not totally clear what limit you're actually even heading towards um so i think that having a good understanding um of uh of of what we should predict for these sorts of things uh will be in you know starting in the quadratic case maybe using that to build up to higher degree cases will be very important for future empirical investigation of the conjectures for class groups of higher degree fields um all right so i want to yeah i'll show a picture a little bit about what some of these um what some of these things look like so here is um a graph okay so there are four um lines uh uh in this graph and so these are um these the and they're three labeled here the three labeled are all um moments so these are moments of imaginary so these are of imaginary quadratic fields and so remember we said that the moments of the class groups uh you know of class groups of imaginary quadratic fields should all be one okay and so this this purple line is the z mod 5z moment the average number of surjections on the z mod 5z and the blue line uh is the z mod 3z moment average number of surjections to z mod 3z and the red line is the z mod 3z cross z mod 3z moment so it's uh and maybe i should explain this um axis here is saying if you take discriminant up to x and you see it's um plotted here the you know log of x over log of 2 so this is like you know up to discriminant of k is less than 2 to the 30 so this is as the discriminant increases you know up to you know 2 to the 5 2 to the 10 2 to the 15 2 to the 30 it looks like this goes up to 2 to the the 32 so as x is going to infinity the purple line and the blue line and the red line are all predicted um to to hit hit one here um and so um even from this picture one can observe many things uh uh one they they all seem to be coming up from below all right so it's not an asymptotic being being sort of approached like this or something they're all uh coming from below which really suggests some kind of secondary term um and that persists with with with you know this is just a sample but with other other moments this um uh hitting one from below persists so suggesting there's some sort of negative uh secondary term it's interesting uh that the here if you see the purple line the z mod 5z moment is a is a head it's getting there faster um which you might be kind of surprised just because it's a bigger group you might it feels like it should be harder to have uh subjections on to z mod 5z so that i really have no um explanation or understanding of of why uh the z mod 5z moment is ahead and then this of course this is this z mod 3z cross z mod 3z moment is is slower not even so clear uh maybe at this point um uh where it where it's going to level off and of course it theoretically this this you know red z mod 3z cross z mod 3z moment has to be behind the blue one because you can't start having subjections to z mod 3z cross z mod 3z until you have subjections to z mod 3z so um there's some theoretical uh restriction uh there uh so maybe yeah i'll mention um yes yes that's what i'm gonna say next good question okay um uh so uh yes so um this uh chart is is from a paper uh that was mostly interested in this other line this yellow and green line and these were being used as a reference to show the kind of what you would expect convergence uh uh to to be like so exactly in the uh in the um sort of uh uh way that i was suggesting uh for the class group distribution conjectures for higher degree fields however this is um this is a case of a generalization not two higher degree fields it's still of quadratic fields um but this is so remember we said that these uh you know z mod 5z moments is like the average number of unrammified z mod 5z extensions and these are the unrammified z mod 3z extensions and so this is about this green and yellow line is about counting unrammified um uh a four uh extensions so that's why this this a four of course um it's that isn't about uh surjections from the class group to to something it's surjections from the fundamental group of of spec okay or the gaol group of the maximal unrammified extension um so it's not directly a class group thing that's what you see here in the title of this paper not a billion coneland straw um uh moments uh and so uh the the context is that it was just it's very helpful when trying to uh to guess where this uh is going to land uh to have to have a sense of the growth of other analogous functions so just in case you're wondering we predict that this is going to land at two um and uh and since you believe that this is going to land at one uh then maybe you can believe this is going to land at two um the the other actually the other i'll just note another thing here in this picture there's the sort of green and the yellow um so uh the green are is a computation of actually all fields up to that discriminant and then the yellow uh is a computation of sampled fields so instead of all fields of you know uh discriminant we randomly sample uh some and if you are actually able to take a good um a good sample you expect that to be to be very good uh now for quadratic fields you probably aren't ever going to i mean it's going to be a while before you need to do that because it's so fast to compute uh class groups and some you know but here um one has to make much more expensive computations and so this is something that i discussed at several places in the notes um that especially uh especially once these computation the computations you want to do um are more expensive per field when they're doing things like computing class groups of higher degree fields you it's probably very much quicker and just as useful to have sampled data where you don't do all the fields up uh to uh discriminate two to the thirty but just have a a good good random sample of them all right so um all right so we do actually understand so one thing theoretically um about this picture which is actually this this z mod three z this blue line this convergence um i said oh it really looks like there's a secondary term there um in actually both cases but we know we we do know one thing um so we're you know the moral of these several talks is but we can count cubic fields that's the thing we could do all right so um there is a known secondary term uh so again it comes from from counting cubic fields really really well so um uh from work of uh barge of a shaker at zimmerman and taniguchi and thorn and then actually a more recent paper of barge of a taniguchi and thorn um we and this is a little bit of abuse of this notation but say for k either imaginary or real quadratic fields this z mod three z moment and the reason this is abusive notation is i said this was a limit but if you forget about the limit with this this average without the limit they prove is one minus and i wrote this minus here so this c is a is a positive constant so this is why it's coming up from below um x to the minus one sixth all right so this is so i was talking about secondary terms and error terms so this is precisely what i mean by a secondary term they give c um you know so c greater than zero is given explicitly and so that is that is explaining this um you know this gap here that it's negative and you see by the time you get to here it's sort of approaching at a very regular rate all right and that's the that's the the x to the minus one sixth rate uh and then beyond that um they have you know a further error term at x to the at x to the minus one third um uh so um yes yes oh except oh sorry for real quadratics um for real quadratics it's one the the value of the moment it's one third and the and the constant this next constant is different too but it's the oh so this this is this is the real yeah the real case all right so we'll say that that is for imaginary quadratics and this is for real quadratics so i haven't talked about you know we didn't talk about the values of the moments those are those are for the real quadratic case they're one over the size of the group and that's that's worked out in detail um in the notes but they have the same uh you know the same shape secondary term also negative just with a different constant um then the error term okay uh yes and so uh just the final thing i want to say is since i'm encouraging a more computational investigation uh in to um you know into the speed of convergence and sort of secondary terms and error terms of these i want to point out um uh one uh one paper so this is of um lewis and williams sort of who did some numerical investigation into secondary terms and this was a um an undergraduate research project it has has some nice data and like if you want to think about this i think you know you you should start and look and and see um and i just pulled up one uh one chart from their paper to highlight so this is the the difference um that they're seeing between a predicted a value and an actual value and these axes here start at zero and so the fact that uh um the fact that the differences are always positive you see even in the five seven equals eleven twenty the fact that these differences are always positive and they're not going um below this line is just more cases of this claim that i i i told you seems to always empirically happen um that um these averages seem to be approached from below uh and also i think it's a very um very interesting question to understand sort of heuristically like um why why that is is so strongly the case um all right that is it for today okay we have