1/4 Lfunctions
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Title 
1/4 Lfunctions

Title of Series  
Number of Parts 
36

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

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01:29
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Infinity
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02:37
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11:28
Multiplication sign
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Kongruenzklassengruppe
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28:22
Standard deviation
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33:52
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39:22
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52:19
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00:02
I'm all for it wrong
00:10
parenting wearing and serve the
00:15
title was vague put them on the talk everything about functions so somebody talk about what some topics related to the value distribution granddaughter moments of functions 7 so too are ready to start with you can imagine so we have some I'd some notion of of a family of 4 of functions and this might mean things like looking at the values of fast say that remains in a punch and they tell similar society where sigma some fixed number and the wary this large sale of city life would be in captivity and twice capital and you can ask the distribution of values so far off the subject
01:30
or we ought to give some of the examples you could take a character kind what Q violence a primitive and ask for the distribution of of else Sigma Chi again for for a fixed value of Sigma and ask what Q from the left Cuba
01:54
at infinity want to give some of the examples you could look at top of a special class of characters which are mainly quadratic characters so these are parametrized by fundamental discriminants state and you can look at objects like the kinetic judicial functions again at some point Sigma and that the various overall fundamental disagreements on some points so the of some kind of samplings of
02:38
sample problems that you could consider and you can look at other variations of this you could take some automorphic fall may be impressed that by my characters so let me give us maybe a few more examples you could look at the alSadr might have where f various let's say over the Home Office I had forms of Quakake will you can vary the level if you like but that's very very the weight and the weight came this supposed to get lost and that in 1 and everything that I say it will assume that the cell function to normalize so that as a function which connects S 2 1 oneliners so half will be the central point with so so or something related to this would be to fix your favorite model form and then do twisted by 20 characters sample objects that we would like to study and then in almost all of these cases but the only thing that we completely understand this saw what happens when passing was bigger than 1 when they're in the range of up Williams so that you can
04:24
say quite a bit like golf like let's say let's say we take ,comma health Sigma Chi which has just given by this summer .period products and then it's easy enough to say something like Well this is at least as large as more Americans than all the Coyotes cost 1 and that most of largest weapons when all of tidy sum minus 1 so varies between 2 constants and can get arbitrarily close to to this constant or arbitrary close that constant by like choosing the 1st few prankster appointments of England so so this is thought you could say easy enough to understand but even even this problem of looking at the values of functions to the to the right of 1 can be that I can be nontrivial if you're interested in Baltimore forms where you don't know that I want and conducted so so even hear some problems will not easy there are some subtleties if we don't know which we don't know in general follicles lesions on his cell functions so maybe I'll just say here that we can say things here but I'm not really going to focus on this but let me just say this would go far military and Shannon which deals with problems of this kind of bonding of functions at the edge of the critical support just a little bit to the right in situations where you don't have that I wanted you to think for example of muscle OK so so the 1st problem which we just nontrivial would be tasked fall in the value distribution of the edge of the critical stroke then we went said 1 so so maybe we just focus on 1 problem here which is of special interest which is a case of pop up and persuade could add a character so so that's especially interesting say that this community is negative and the size of the Osaka less than excellent then we would be interested in I L 1 tiny since we know that not that multiplied by about this sequel to the task unless he has been a constant electrified divided by W which is the number of of unity in the field was W is usually too to up to some factor of like this gives information about the customer of appealed to square of suit asking for the distribution of letters of 1 carries the same problem was asking for sir so that's a problem that we don't understand very well unconditionally the only kind of bonds that we know of that at 1 time is bounded by by so this is so fully expect that you can put in a small slightly smaller constant here as well and this stopped this season and then we have a long bond that it goes like this at least likely minus epsilon which which is inevitable and remains an important opened but this is not the truth of what happens 401 K but if you know or something like that General Instrument Watson then we have much but bounce so we know that has found that most by some constant times long loyalty and some other constant times so how should 1 culture will think about this and also you know however these large and small values come about and usually what societal alone OK so this to think of what the size of the 1 kind he should be we should go back to the south of the 2 real example that I didn't seen was bigger than 1 I can just write it as lot quality which wouldn't and then get a prolonged so you could ask Is there something that's going to say household will have to go before I can approximate Katie might follow but best so the will be something depending upon the deed you should think of concerned about sites and the will of course but will maybe good the S X goes findings will certainly need something of this site and you you're interested in making easy as small as possible and there is also being true and unconditionally would have to take Zebari worry lot like may be larger than acts at
11:29
any rate may be even larger than that may be likely to the logic square of something like that but does but is if the delays in airports is true then 1 can take Zita fairly small flight logic square and if you if you once you want to know about that these bonds follow because this product at most can be as large as a product of 1 minus 1 will peak in up to logic square which gives to the upper bound and it's at least as large as this is for and and this principle is related to some of the principles that we have like once you solve problems that we have like what is the least sclerotic residue on 1 messages so so find the smallest value of people which Canadian appears want or try to use the best minus 1 and that is a problem for which unconditionally we don't have very good results because they the least time assistance and power of the like couple maybe a digital service it will get done he is less than the 2 1 authority here maybe only that the assessment in the 1 the characters will it's just that but the thing that pulses with the loser toward these cases you can get Can you can get positive made values once a size log no no be the 1st thing to to get a startled think you want these value distributions would be to understand problems of this type like this these Connecticut your those knew what should be the troops generational to something like long Klawock square but that's not really the truth please the Truman the truth in these cases should be that should be that there exist such a value really that you can't take anything which is a little bit larger than logic so that people politics Exelon but steps along but you'll see that I can make it even a little bit more precisely what extent of under the reason is simply but the thing flat the fact that the so enterprise it could be possible 1 0 minus 1 and it's roughly plus 1 0 minus 1 with probity onehalf so I can't think of this kind of P as being like a random variable and this random variable takes values roughly 1 minus 1 with people probity but you might want to be a little bit careful because those at the possibility that the value could be 0 when divides the let's allow for that possibility and then you can work out the right product properties here of this is with probity appeal to twotime people want this is probably the 1 old people want and this will probably appeal so this is approximately 1 albeit 1 or more people 1 because these conditions the square feet the pressure well we just pretend that his plus minus 1 with equal probability them to get zeal numbers pointing in the same direction the probity this time for all of disease this is about 1 or 2 to the piracy and antigen imagined that if this problem is much smaller 1 over at the maybe you don't get any numbers which have the property of stop when the so that's a plausible conjecture for what we should expect that for this Connecticut studio monitors you problem and you might also think that that's a possible conflict what the values of a 1 Heidi should be that we should be able to take this oil products and take the oil products up essentially logic logic and 1 % along and that's a good approximation to the value of loan I think so by the way this conjecture is stronger than insensitively predicted values which are operate opera bonds fall and 1 Canadian law 1 from a committee which are about half the size of In the case of that sort please that's completely open but that gives you a good 1st model for how to think about the values version of I Want Candy we could say well the should be modeled by looking at the random where the XP so they can be independently of 4 different crimes and satisfy the size of this being bused minus 1 with equal probability so I have an infant product here but you can check that this included comic book on commercial almost surely and the reason why convergence as well as the convergence of this product is related to the convergence of some of the former XP if I think of all the of the site under the convergence of the summer's OK because if you think of the XP their past minus1 1 equally often so counting something XL them will be a square root cancelation of some of the expertise and the weighted down with something more than square which will give a convergence of this convergence formal show so this model has been studied in the context of understanding Katie for us for a long time but maybe it goes back to work off motivation and trouble and also earlier in the 70's and they proved that the distance that I Want Candy has a nice distribution of a smooth distribution functions so you can compute the probability that I want Katie is bigger than templates
19:56
of the probity that is less than about 1 over 100 and up and sometime back maybe 10 years back now gamble studied this carefully trying to determine whether what uniformity we can match the distribution of ideas about 1 tied in with the distribution of these random 1 of so we could prove something like well so you the way you do it is by computing the number of fundamental disagreement about the size Voegele 1 Kiley's bigger than some number which we normalize neatly Gamal time we began with 4 this month and I'm concerned that comes from this point and you would like to figure out when this is approximately given by the number of 5 the random or the product to it's called acts being bigger than easily gamut times the list of IT away by the number of contents of and we proved that these 2 objects moralist matched in somebody uniform range but is true the towel up to something like low below decks blast maybe so so it was the son Commissioner it's a lot for this for a long long long and that and then if you assume injuried then you can replace the slope for warehouse OK so what does this mean well you can ask well what exactly does the probity of L without all this 1 Katie being large was approved pledges of it some crazy function is not something nice like calcium and power anything like that it actually behaves very strangely it behaves it decays W. exponentially so this behaves like I need to read mind some constant find in the tower which is still legible very quickly and up to this asymptotic not known exactly quality on the phone I wrote it down because it's mine . 8 1 8 7 something like that and there's this constant that appears in the exponent is this some weird things so I don't to write this down just illustrated this is not some universal distribution it's something that you can compute in this case you get some answers if you computer for someone it doesn't have anything to do with just the the family of functions test things to do with the actual versions of what the Europe ranges over and so on OK so so what does this mean if you if you look at this problem the because it's doubly exponential it means that once towers on size of access than this probably become something less than 1 over and because it's being developed by the style tell really has to be if on the scale of Law blog exboss Loblaw blog and then the then this proportion becomes less than 1 over and you expect nothing be so you don't expect this equality to hold 1 style is bigger than this plus this past 20 when there should be nothing so essentially that's a very wide range of matters of style in which this inequality can possibly hold and we have a secure and the least on generator that it almost holds him that and then in which account OK the fact that the solutions for such a wide range might lead you to believe that so therefore we may believe that the system this conjecture but that be of the that which again as I said you know it's itself in some sense beyond beyond our so if this is true then we kind of thought you could say that we completely understand the distribution of ideas about 1 guy he is be it appears like an underwater quality and it seems to behave like an animal products in essentially the whole range and when will you see a value of well of course the chances of a that it's just lies between . 1 and 10 University of value which is not in of strength but this is not the only question that you can ask about up what values even at the edge of the critical struck so that we ask you 1 more question along these lines along which Renault extremely little so let's not let me just think about the negative discrimination I would say Look at this which basically same not minus 4 Oakland Street now I know this of course is in India which means that these things are not arbitrary alone was they have to lie in certain markets near integers divided by sports events and in the interests of the and into the like if I taken in development 1 lowering taxes liked understand the distribution of vulgarity in such a short you know this is of course impossible because the random model is not seeing anything without medical these possible I think so so let me give you 1 convicted him we would certainly expect that but every number is the class number often measuring Connecticut field I think this is an obvious connection and the reason why the it's obvious is that you know what makes this commands of precise you take that possible was
28:23
they all sizable exploited the fact that they conspired and here you have X number of snapping don't discourage effects numbers to each number should get its fair share of thought of fields for which at a cost work so there should be enough legal texts so if I give you a number given a number 8 there should be about hello age fields it was also at the discussions go up to a square among the chance of you landing exactly on age might be like 1 over so what this is forgetting something like Well for example rages on then due history tells you that the discriminants for which you can you can have a few with customer age they have to be prime so maybe it's not actually aged babies like H along that and if it is confirmed as do biological off to them you know if a distance away a lot more often than the discontent readers will buy a large number of crimes and maybe there are more disturbances of it's so certainly I would think that if if I let this number will be demoted by Apple for each the number of deals with customer then I would expect that maybe this is always bonded by something like maybe age times luggage and maybe bounded below by something like 8 or maybe 1 can make more precise contemptuous of although so far 2nd that nobody has a good doesn't it gets somewhat the politics it should be what we all we know very little about this all we know is that and that Paulson this work on on the distribution of that I've just been talking about creating computed the average of function of a page that turns out to be a fairly nice constant but the was remarkably weak you know I can save the square root of luggage and I have no idea of how to save anything more than that like maybe 1 8 square will the attend and because you can get some ice and other forms of its American you can also prove something like the number of fields with often ranges of mostly squared some powerful obligation for all it this is very weak just all it says is that if you look at all the time all the fields with custom work up the gun all basically accumulate on 1 value or maybe on longrange by 1 the rules out some things that says that you know it's not up so I don't quite know how to put my quantifiers here but now I think that say this is all that's home to the not almost all of fields have class people to a power of 2 kinds bounded but if you ask for if you ask for the same question with so I hope it's clear what this means is that there is some also density of feels like this is er density of fields with customers are a power of 2 types of all along that you can prove but I don't know how to prove that when they're all the same question whether customers was can simply be a pile them but power 3 times abandonment for if you ask me what you know 2 3 5 7 minutes wide open to figure out how to so that's so very wary this is you I don't want to get so there there are some you know there are fluctuations in that saved 3 Dwight's across the world feet away age then that seems to bump up the Rachel but so so there are deviations that you see by looking at the tables 11 were exactly how old it has been so to we have a right to you then have to
33:54
assume that they're all independent of each other on the make some formation and then you have to put in the G theory things fall policy so I wrote down somewhere and a version of this with some policy with some with the data were luggage and a powerful and there should be on that or that I feel kind of confident about what I don't feel comfortable with the consul :colon so so this is a very strange that these values of 1 tied maybe you know we know they can lift some nice with distribution function but it could still be a worry Daniel kind of distribution but at any point you find that they accumulate in body short intervals on such a small number of and we don't cultural things like that sir thank you so that's a good way to ensure that something is a little so that's what happened for the end of the critical through you can ask for any other values Sigma which is bigger than unless someone if you fix signaled have then you there has been some work on this recently due to low Missouri it will also discuss things like that's a good look at the analog of what I was talking about and that is a detention cases if you could they don't want you can think of how this is distributed In the cop explained so it's real and imaginary size and its amplitude and its argument if you like and he also has extensions of up to values of Sigmund applied strictly bigger and have and the story is more complicated but you can but kind of similar in spirit that you can still understand things pretty well using their animals To Hell itself so what makes this so that's a for Sigma Chi you would analyze it by looking at this 1 minus 60 it the by most and what makes another model workers that again if you think of that convergence of and we look at Expedia will pitch the signal so again at if start canceling out the square root of cancelation because their plus minus 1 with equal frequency you see that the sun still convergence of supplies bigger just almost but so secure 1 kind of quality of thought of this work but if you if you fix said no less than 1 and if you look at the value of fire this is not true someone legislators that the books to care is close to any given real number positive real number for that in many ways of indiscriminately before there's a distribution function for this and any given real number around that there's some positive density with which you take values and that this should so the reason why I was going to write something down I hesitated because that would of course love to be able to prove that they would consist only of positive real but we don't know how to prove that because of B some part of the also to say that there are no real arose out of these functions what because even the the characters for which you might have a 0 that's said signal those are the Institute results which tell you that that happened braved but sadly OK but these values can take any value any given positive numbers about value or
39:24
dance in the set of positive realignment OK so that tells you what happens to the to right of the critical line and now for the rest of the time I'm going to talk only about what happens on the critical of the now we can do discussed you simply have glitches in some ways the Mohandas K. so you can see that top 1 way in which things failed as the random models longer but meaningful if I I don't random defend them on part of this type than these this series does not have become infected by would almost surely so the some of the text field in office of the In the legislation almost surely that doesn't quite and you can see it in other ways too because there was a loss of functions on behalf wine and when you have such a 0 it doesn't make sense to approximate by any kind of weather problems here OK and you can and it's also reflected the fact that some very basic questions about the value they don't have a slightly Our unanswered so here's conjecture which goes up to Ramachandran but the values of the Gulf have signed the innocently from this these values dance in the country and this is this missus open but that may be it's not so hopeless that that and maybe somebody will hold on to this work on this by by Emanuel calls the conduct and naked volume connecting this to 2 connections falls at OK this is this is in in some ways an aside I'm not going to talk much about the size of this problem so 2 1 difference is that while these values to the right of the critical I'm actually have a distribution the values on the on the critical line don't have a value distribution what you have is that is a different kind of results which is due to somebody and we like the final lots of for for L functions which we don't have that last year areas and that's the key to 20 so you should take the along with of function and take it's a real part automatic punk so if it's a 0 than the real part stopped Of the law would be same minds included but it happens on a set of measures are also doesn't affect any of the calculations the values of this part of the plot of land distributed like and calcium seem to like a normal random variables so the approximately normal which means 0 and variance about half of global OK I'll explain the summer this year in a little bit more details tomorrow and also what it means for moments but for the moment we just say that I know what this means is that OK this variance is growing so so what it means is that if you take a value of laidoff hapless idea and you just look at it there you that the 2 cases even at some large number inside or at some small numbers and you know so the probity that you see evaluative of size and it's going to be so so therefore Yukon you can't make progress on this conviction that the values are dense and c because that is actually of a set of which has aromas look that's when I can get to the main topic of the and the customer next which is a classical thing that goes back to 2 Hardy in forward which is to try to understand so their ideals to understand the values we can offer of opposition function by studying the moments of
46:42
severe case so case let's say some natural number or you could also just take a to be some real number which is positive maybe and what it also makes sense to consider complex moments as well in fact for the supplication tarantulas connecting need to I understand something complex problems of Satan and 1 reason why they were interested in those edits were easy to see this is just the to give more of the zeta function of course if you can understand this fall large values of faith you can say something about the normal was a function which is a little of it was so there's been a lot of work on this side on this on this question well from the 1920's due to in the world but so far in this in this case we saw how long the asymptotic formalism 2 cases suffocate if 1 intake of students so fucking equals 1 this accessibility to the lofty which is the Italian forward complicate close to so maybe I'll like that it is anticipating some other colleges that entitles us to transform foreplay and that was due to the meeting so actually inconspicuous .period this quite interesting because he considers he considers more general objects like a State City summit that may not quite be hesitation with his office considering the 4th moment you also opportunities for wearables if he said the mall equal to 0 then you get the full amount of data with he works of the asymptotic following the surrounds the saying that it's more transparent to see what the shape of the asymptotic formulas in when you when you disentangle the variables like Co so for a long time these for all the good that was known and that it was not even clear what the right right for these moments the people get the right exponent right so so also also folklore conjecture but I don't quite know what contributed to maybe maybe Titchmarsh from various local citizens tribute to those of local connector that the than that to get moment that may give us the name just to stop writing it all times that this up this has emboldened some constant times the looked with a square 2 and them maybe in the early eighties also sedition Connery and good but you should not let this constancy factories nicely as the product of 2 Constance A. K. M and Jake and that we're well OK at the moment of this not so profound to write this down what the point was that there is that there's a natural thing that you could associate with the with the that moment of and with a Cape a function that has to do with series which is the key devise a function decay and warrant for me at least in the range of wouldn't no you could you could understand it could think of this this expanding this moment as being like that in the main square of the power and think of using a possible that on so a possible argument would suggest that maybe the asymptotic should depend on some object which may be looks like those became square where your on behalf Lancelot cement the hapless lately and that a few cities
52:21
if you take the square so that he could decay and square where and that what is bogus about this as well as truncated in choosing to truncated and up to which is not motivated by anything at all except that that's a natural scale may be on which this this this then you can show that that this object it's were easy to get but a formula for under the terms of BS about it because a came times to look into the case so this is a natural order on which the moments of their function should be here and then this G case measuring some kind of deviation from From what you would expect very naive and in this notation that hardy little wooden income results favored the 1 is 102 to and then the other values of decay were not obvious for a long time and then in the 90's comedy and conjecture that the trees 42 and ,comma and Peconic conviction that she forest where controls and 24 and roughly around that maybe it exactly the same time that they conjectured this heating and made from the general manager which critics want the case should be at the college's quite nice and it's a little slow condition of thinking actually it I said this for but you could actually make it makes sense of a Ford Mondeo values as well you simply have to be spy gamma functions and you basically get something which was restricted to advance double gamble yes so that's why the Barnstable gamification congressional delegation moreover so this is the case all of function but there are other families of El functions that you could consider as well and you consider moment in these families so limit you to 1 the examples which are kind of typical you can look at all the components decides acts the fundamental and up from the average of l have kindly to the case so it may a case some natural number these values are all expected to be nonnegative but we don't know how to prove that it'll be a important results do you also once when I want tidy but what anyway they should be nonnegative few willing to assume that he also the rules arose between half 1 and up and here the convict killings snake and it just as a little bit different this is the this some constant the kinder toward me analogs of Keating's state some kind of some constancy which can also be specified nicely in terms of pictorials but which I don't remember offhand at times but what's interesting is that you get a different Pollaro long Xia NYTimes gave 1 over 2 and them to give 1 more example let's see you fix your favorite modular format and has taken out a crossover and he gets some constancy K may different constant from here it didn't so maybe in the last 5 months that me just tell you about what I wanted In rest of the lectures it is a the am so you would get the well was not clear how exactly would formally that you get the right color and that's something that little of what he would do so so so extreme should predict that you should get some constant plans fealty to his car and out and there will also be the key to trying to think about what the what the distribution of these should be that we would like analogs of some of 0 in these candies ,comma which which we don't want to look at the 1st point is what factory just said don't try to make a Alex click the link between moments and this body distribution things like some and expected analogs for world functions but maybe this 1 come but so many of the things that I want explained this to the also figure out something about where the convictions for moments come from so there's particularly my way deformities convictions which as you do quandary farmer heating Rubin state on the heuristic is is very simple and it gives a very elegant answer for what all the momentum in all the families you can think of should be but on the other hand that was so so you can leave and this has been verified but in many smaller many small cases which held mentioned some of the small cases of moral but the reports maybe unsatisfying is that there is a very nice expert when conjecture with all the proofs in which we can check the conditions for all cases are wary the very unsatisfactory and that they involve you you have a huge mass the new check somehow that the mess that you get matches up with the cell the size of the it's not very illuminating the process and then we have a general techniques which allow us to give law for for all time moments so acknowledged if you know some moment and you have a little bit despair when you can get along with the right order of magnitude for all and the 4th 1 would be that a complementary principle to those that if you have a bond for some moments then that implies automatically that there are other for all small moments so of course this is all this is obvious don't get the right color flown by holders equality so that the point is you can get the right moral law In what these cases and the last 1 is related to this kind of argument now we know that on your age you can originally approved after months of like order of magnitude the money didn't essentially any case that you can think of stopped yeah so I'll explain that the only and that more than GAZ the ad last month I was in the heart of the freedom went on what you using the years but there would be more of the same looked into the case cracked from In a few worry the family than you can still get what the answer is going to be like this 1 the result India's but there you go through the book from you