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2/4 L-function

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2/4 L-function
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Logical constant Functional (mathematics) Group action Multiplication sign Real number Range (statistics) Routing Parameter (computer programming) Inequality (mathematics) Rule of inference Power (physics) Goodness of fit Sign (mathematics) Pi Root Lecture/Conference Meeting/Interview Square number Integer Area Theory of relativity Moment (mathematics) Approximation Numerical analysis Calculation Summierbarkeit Right angle
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I so over time and we will you
and so be recalled where I ended last time that I began discussing the problem fall moments of functions on the on the critical line so for 4 hours later that these conductors of people with complete moment behaves like a some constant times he looked into the case square and the CK was factors as the product of 2 relatively well understood things case where easy-to-understand it can be written explicitly in terms of on oil product can be given some of the oil products which commodious intermittently on this GK was this mysterious factor which I rode on a formula for conjectural formula for lost time and I also mentioned that you could look at analogs of these in other family of the families of those functions so for example if you could look at Cottonwood queues so these all conjectures lets a few average from the characters and they should be as important to some other constancy Cape Kewell cute look square but let's accuse Prime Select don't have to worry about how many primitive character there 4 for a few look at it fundamentalist components and you can put characters then this should be a symbolic to some constant times and if you make a cure mentioned so sorry that should be case square of course but if you make a cure a dual-mode 1 more value of k on May the so I won't mention results in them and that and you could look at this so far 4 modular form where you expect something like exotic explicate landscape and so 1st let me explain where these conjectures come from and then I'll explain what we know afterward so the you focus will the set present the wall fire you know of many other kind of conditions that you could write them so why do we have these shapes of grief of thought Of the substantive it well soon there's also a connection between these moment conjectures and the distribution of the Roosevelt functions animated series so so let me just say that's for the zeta function for states that if you look at his and suppose we assume the remark boxes and Michael's arose as hapless gamma and diaries these garments in ascending order then there's the conjecture which goes back to To Montgomerie and with numeric supply will the scope and that and then extensions by Republicans ,comma that if you look at the surface of the sea zeros get closer and closer to each other as well as as as you proceed up the up the critical strip critical line so the end 0 is roughly of size To provide and overlong again because there are about the looked Israel's to the so you can normalize this by looking at top gamelan time slogan when divide but comprise so the and normalizes is about size and and you can ask for the spacings between GAM and so Terry mentioned that for Prime's if you take the normalized spacings between crimes they should behave like spacings between random strong means facing 1 this 1 is expected to be the front the spacing between this Ordinance should correspond it expected to cover the 2 correspondents the spacings between the eigenvalues of large random agencies the so maybe to give up to give 1 1 kind of ensemble that you could take care of you could look at that you could look at the unitary group and choose a matrix uniformly from Palm measure on this group so and then it'll have like values which look like affiliated I want you to the idea and these numbers they
don't want to pay and lie between legitimate let's say rearrange them so that they lie between 0 and 2 5 and that's what your interests and would be this than normal a spacings between outdated G 8 times and all so then the difference between 2 consecutive eigenvalues would be if you normalize the site angles the normal spacing will be 1 and that water was facing that you get should correspond to the space analysts for each matrix you look at the dispensingDistribution here and then you have resettled overall matrices and the school you so when you average it there some measure on how how these they don't want to pay chosen the measure is not simply the data won't be there and it's something that keeps track of listening to the I think today and you go from 1 to an unknown claims the on to suit the different skills from just taking angles independent of each other is that the angles here repel each other and you don't want to to end of spaghetti close to each of the space of the space of the group went any and somewhere close to each other wasteful small so that's the kind that on on the 0 so far all those functions and the kind it just says that you should look at the spacing between 0 so far random agencies and you want larger and of the Saudi and you let out this parameter capital and good information so Keating and Snape had had the idea that you should that make this event but more precise and you can make it as natural model but would you say that if you look at statistical properties of zeros Fahrenheit then that should correspond to so said you don't just want to say that as the ghost included co-sponsor something on the domestic side and goes infinity you want some relation between PNM on the realization that they that they are that the postulated was that 1 the spacings between 2 convicted of zeroes here is roughly 2 fireworks lofty and spacings me to come together like letters here is softly to pile and so that kind of gives you the sort mystical instituted a stake said these 2 things could each other we should take and to be roughly speaking on the scale of laughter and you look at random matrices of size but so if you if you believe this then that gives you a way of thinking about other problems on functions so for example if you want to think about the properties of the adult hapless the on-site day their own fate capsule the than you might expect that this is modeled by by some other objects by the the here who is of the eigenvalues of your random matrix and that would be the counter-cyclical around matrix size and which is about what to do and then you can imagine that maybe the conjecture
fault that took its moment of laid off half IT should correspond to what we can find fault the average over the unitary group off the characteristic polynomial "quotation mark some metrics chief however it will agree to it so so now decide 1 can calculate exactly and in fact was done by was something which was computed by solo In a famous people that he wrote in the Norwegian School general roundup and so is the he computer that this House Apollo has it's a polynomial but for from this would follow that it's a polynomial in and of degree case square OK and that so and this should correspond to lock the decay squared so you can see this matching and leading all rest bullets for those involved is constant GK fines and to the case and this is the motivation for the Keating so that this summer in this analogy you don't see this as a critic factor AK which it said was easy to understand and you could light and consultant on a par at the moment Prince of course thermal prints on the side of offered under metrics calculation you can form an analog of this in all the other cases that I wrote down the group that your average out might change depending on the family that Europe that you're interested and you can form analogs of the some of these conditions so so the other can thank you for what he has been in the US but you can commemorative well let's say just answer with it doesn't matter it would make a saying the key aspect it makes no difference going about it in 1 of these other families who might have evaluated that if there is because there's no special .period the other conductors of formative similarly but you would so soon so that the would refinement of this conjecture of Montgomerie and suffered examined by work of cats and so on and on families and even some family of functions that cats and sonic prescribing a symmetry group which kind of tells you what family of random agencies you should look at order to understand the spacing zeros spacings all but l values and the family and you use that the model the moments as well and that and and it gives to these different predictions and seemed to work but the seller new should visceral 3 cases unitary which is where examples are likely to function and the aspect all the failed entrance or you can look at the start of formal and here the typical examples would be like with his cell phone of modular form and you see this feature of the timescale minus 1 of 2 coming in this example and the last would be an example would be some the games where the prototypical example of something like just but think the failed functions but so this is 1 way of approaching the size of these moment connections that it's worried that's where I pretty much in you know you kind of telling you conceptually where constant GK should come from what it also may be in 1 part of it which is unsatisfying is that you have to put in the same case by hand but there's no way in which you see that arising from the so maybe 1 1 heuristic along these lines should be that maybe we should think of the zeta function as having 2 parts 1 is that it has said it looks like a model product up to some point on them this summer but which comes from the your McCormick 1 best separately and must match something from the AK times some some islanders think on the average of 0 Park should give you something which just use this animated calculations so this work along those lines by USA the comic In the kids the back I'm not sure that I'm remembering the names credit so maybe maybe former Manitoba not to look at that I think
it might be far more of his but it we can give jockeys because were they have been so wave formulate some hybrid models of trying to write about L functions in terms of both crimes of some .period plus some contribution from the worlds of any can study each 1 of these is 1 the separately and that that produces the best contribution from the prince produce the AK and the controversial look 0 so produces the cheek focus further so now I want to tell you a different way of approaching the sort of approaching this conjectures which is due to the contrary far more Keating which for is just going to do moments so this argument actually will fall real moments as well and for integer moments there's a conjecture which will identify it's a lesser 1 advantage of this is that it identifies all the law overtones in the woman connection 3rd so for example in the case of the Kate took a moment of their function is produce that there's a polynomial PKK in Lord of the so what degree he squared up to identify for you although all the terms in the expansion for of all the coefficients of what is a very messy polynomial it would it won't just be this polynomial because as you can see it also have include political factions corresponding to generalizations of the thing and this connector is also very worried simple and innocent and so I want to explain How about how this can toward so let's say less I'm going to do this in a special case and they're out we look inward look at all the characters kind what Q say the supplemental assume accuse Prime View life and something larger and I want to make a conductor for the sum of l have ties to the and the idea here goes back so I mentioned last time that a new computer default moment but he computer the moment over their function with various ship but with 4 with 4 parameters for the idea here is to do something similar here not just to look at this as good as it took a moment of 1 object but introduce ought to get parameters so it's convenient if I look at the completed function instead so let's say we multiply this by although some epsilon so to complete deal function by multiplying propaganda from gamma so this is either 0 or 1 depending on whether collectors the characters or even or even or odd and this is a gal some was signed 1 the absolute values OK well wouldn't worry too much about this so suppose instead of looking at El have tried to to carry look at we look at the falling object instead the so introduced took a new variables out for of KMB 1 debated in and that at times of course if I set all the outposts on the basis to be equal to 0 then I couple of the took a moment that I'm interested there are some factors of the sort of these guys but of course I can just removed from at the note OK so so this all that water would then you can see that the tests but that I'm allowed to commute some on the answer should not change so I can't Bermuda when all the health and that clearly nothing changes I can also commute all the bankers and that nothing changes but also I claim that I can take but I can take this said put out people want to be Peter decay they don't want L cables to be better to unlike claimed that if I actually I can think of from
Udinese to care variables and there's still nothing should change and this is a step that you have to check on the reason for that is that if I prefer make a additional all these 2 convertibles than some other parcels will become mindful that say some helpful 1 becomes beta 1 and beta 1 was backed up for 1 if we do that then I'm using the functionally question twice once to replace this fire have minus 1 on using the function aggression once to the places by have be 1 kind and if we use twice the sign of the functionally questions will cancel out of Swat Epsilon Chi transatlantic I OK so nothing changes you can also what we see is that there's a slightly nontrivial relations that I can commute the Alpha One to health care after 1 to health care in this notation on them again nothing should change so the water answer that I have the water around so that I want to make this and so should be some the action of a stupid what by the OK so now let's make a guess as to the as to what the answer should be will make a very naive guests will expand all these into this usually series even those nothing might actually come work on them just identify died so um so we have various factors coming from so you have these Kewell apply fight to the hospital 28 so I don't forget this day that appears to so this is just a nuisance factor was not worry too much about this and that's what I have is the sum over kind what Q and then I'm going to expand out all of these the cell functions as usually series and not worry very much about convergence so I have a sum over the K variables and 110 K and the caramels and want K the memory something like something like that OK so now we do 1 more piece which is not justified we take the sum of all the characters kind 1 not more acute and we use the motivational altar ventilation will give you that the product of and 110 K should be congruent the product of an 120 what you for this for the sum to contribute and didn't to think of that altered legislation that is picking out the diagonal and want equals and 1 I OK so so the guests from from something that might be there so the number of characters times what we you have here times diagnosed well this should also be co-prime to cure few like variables "quotation mark to Q and then you have no need but no note now this object this is something which is nice and multiplicative if I look at a prime it must stop Europe and the same number of times a little on the left-hand side as on the right-hand side on the left-hand side that savored the white 1 fact they might divide some you know it might appear as Peter the hapless Alpha 7 and might appear as some people have minor Spadea 11 French so you can write this as as a product for this 1 conformally think of as a product all kinds speed of various expressions which would look like you know let's say they made can appear in the product more than once but did say that comes with Gupta ones would look like 1 or more people but 1 plus Alpha Jr mine made a key beta held under some or all J "quotation mark 2 and you can see that became the highlight overtones will involve larger power source of so that the Gulf of wood in the biggest being relatively close to the solidify along peace Square that Newsweek emergency series so the main contribution the summer is the speed of the one-plus averaging minus Beta which I can think of as a golf 1 because of a demonstrator so so we would compare With the product overall J & L going from 1 to is a 1 plus of forgery miners with L and then what is left over will be something an elevator in a wider range OK sir OK so so that's all that's going to be my guest for what happens if I just blindly expand the series here and then just adjusted myself to the diagonal so I would I would make it out make a conjecture that the answer is something given by this part of data functions and some other function which is nice and can can be understood easily and this is roughly of the right shape because if you made average in the Bedales being smaller than this is like having got a pool of water case square fault a function no but it but now if you look at the sensor Vicenza is not really symmetry in in the
very so you can you're allowed to commute the effigies that's OK because Bermuda-based tells us OK but you cannot now make some of the outflows become betas and some will be just become so you miss some symmetry but the original answer must satisfy so the marketers so now the idea is now that's up so we are so this offensive against its symmetry escape process escape and what not system so the the conjectural thought ,comma reform was hitting inspections its history simple you just energize this answer so tries again so you take a song or all permutations In this caution and just the guests let's call that the you'll out for 1 to have to K and when you do that you apply the limitations 2 can I so seeing each each time in the gas just if you just look at the this has a whole bunch of singularities it has similarities when world the averages equals Obaid Al on 1 of the Alpha happens to be 0 so this this'll lot of singularities of that expression what if you somewhat tries that than the the object is actually regular In all the other in base so this objects this many singularities but the summit address on serve is centered In southwestern although the cemetery's cancel out and when you get powers of log will you can see that you have the powers of Q will vary In this in the sunset you might have accused the Alpha One Plus that you have witnessed some of the outpost minus some of the biggest 1 in 1 situation and maybe ghosts that that exponent will get flipped around in other cases so so the powers of law will come from France I on the power of Q that you have an OK and then finally if you take if you take all the variables sequel to the world a this it matches exactly the Keating state conviction so I think that the and so this use the functional agreed of Will you mean the use of the word the right but that's not only takes that gives you so that the Council 1 competition where you replace all the offers by Beijing and all the Beatles by but this year because this would be a counterfeit right lipped out but it's the same with the same idea so in a certain sense this moment convicted the simplest conductivity ,comma right down which is likely to be true you you must have the best took summitry and this is simple subject which has sir so very this oriented conductors somewhere well but what what we're missing in in all the ways that promote to attack these problems we stop with the Emanuel saying with something like approximate question we start with an expression for neural function which is you put all these variables you start with 2 terms 1 of which is so want would be symmetric and apply being the the identity accommodation but that's what you have and the other 1 being the from edition where all the officer replaced by all the way to do that than an allusion that and you miss all the other cemeteries and so we when we have to use so you have to make a long competition which eventually become worse all the symmetries that you had but you don't know how to do that we don't want to do that in any straightforwardly we don't know how to keep the cemetery on the staff so what we know about that about this conjecture well so we have a lot of Karen's fault for small values of cases so for a Zeta we only know for cables to sorry cake 1 and cables to the 2nd and the 4th moments if you look at the family of aquatic "quotation mark editor functions really well maybe that we do have 2 four-time on cue so this is kind of exactly analogous to the zeta function but already have some features here which are harder than for later and we now know that equals 1 and came close to look for cables to fall for the fulfillment of a zeta function for a long time we know that there that you can get a message formula with the power-saving Indiana Jones said in this case to get precise information on seems quite quite complicated and only known fairly decently
by work of Matt Young and in the case was accused Friday then 1 can look at the top moment so far but editors sale functions so here we know In the 1st 3 moments period you can take up just it'll have to be even so we know the 1st 3 moments of the seller may be the 1st to argue due to utilize the report is due to meet and then the 4th 1 it's something that is very much on the edge of the looks like 1 should be able to prove the 4th moment there's some nice result of Fahey's Brown his lots of floral characters which implies that if you look at the the 4th moment you can get a bond which is which is almost sure you can get the right color reflects lump of so in situations where you can get a sharp opera bond for the moment you might hope that you will be able to define a togetherness and public so this is 1 such example but we don't know how to find it again public and a little construction so that on jury on the Net young and I you can get 1 can get on the topic actually this assault not strictly speaking correct so maybe that we put them in court we have a method which we worked out in the front and in a different case which would extend to give us some of it from this case but nobody has written down on the other hand thank you use critical supposedly to write here so example where we know and where locals to take the suffer model form the the 1st of 6 the and in this case we know only that the 1st moment 1 which was not about so it's not strictly speaking ,comma but maybe you know but in principle this was done by the Maltese and on free of cost field they dealt with their little with their little obtaining the 1st moment in this case but at the same method that would work for just the 2nd 1 and that knowing this book equals 2 it would also be an interesting In extreme cases we we are almost there the same result of his Brown would give you the right up about all of this moment if give your next to the 1 plus epsilon belong here and again we don't quite know how to do this but on the other so this actually is what I cannot and I proved and it wasn't so this doesn't because stuff it doesn't verified the actual polynomial that you're supposed to see it only points a leading all the time .period and you can't it was a lot of well no other kind of results which have been done recently this would farther Connery wanted from myself and this is a question on what happens if you average Q as well so so you might you would like to prove something more work where 1 can see this kind of body do that appears in the moment connected and we would like to see that this is 42 times something so so the cube characters of an average accuse beacuse square characters the Paul blocks should be a lot the mine and and so the system come follow a case or late fees average and that we would like to be able to prove exactly this result as stated we don't quite do that but we do something pretty close to the so let me be placed by this is the complete function again the hapless Iwai underlying need to introduce integral here the 1st of all I want them all to prove sorry without any way that because it's a complete mental function so the computer functions began a factor will decrease exponentially as White gets lot so that you could think of this as a small Labrador feel function and development 1 but under this week can verify that Anderson told a formula for this in fact we what the crew was something more precise we actually get the full asymptotic expansions that's conviction by keeping a By Connie former Keating science made out with America which is something like development .period so you get a power-saving in the in remainder and then there is some worry might work of fire shunned from maybe a couple of years back where they prove the analog of this would be a moment the main thing is that they see the 24 thousand 24 medals connection so what this result you don't get a
power-saving there in amended this is really just checking the asymptotic you don't know the full asymptotic expansions in this situation and also this and so on cherish but this is so maybe this is the largest moment for which we can check that's convictions no so those of a special results from around and that's what we'll do in the next couple of lectures is to discuss about failing we might want to get good opera law bonds for these problems so so I'll discuss now the law Bunsen makes such as discussed above for these forms so those on the way maybe there the same 1 1 line summary of this is that is that we can we can get the right upper bound by law bonds for essentially any family in which we can compete and do something so so when will weaken computer at some moment so usually like to compute this may be the smallest moment that you can 1st moments like if you if you like the the 1st moment of blast Epsilon which I'll explain what this means is that you should be able to put a little bit of the offer of a difficult moment well and also be able to compete at the moment than this will apply the current law apart from constant for all larger moments so let me try to explain how this saw how this works and this is something which shop which allow was doubled by Republican made and that and with the recent data extension by Maxine which really said the results like this for the zeta function our our our guard classical they go back to committee formed to teach Titchmarsh but you can get along bond calls data have decided to the to which is all right and this is you can you can think of this as just being an application of vessels inequality so you can love you can look at they the case it becomes a complicated object when case size large but at least you can understand how would you know it's in product but that with functions like and the IT you can understand that for small values of caring for small and not take the one-liners and so on and then you you can use vessels inequality to say that if you understand the singer of products the mean square alternate office to the cake is bounded below by basically the averages of peace the Swedes quest but this proved doesn't stop by itself extend nicely too 2 families of functions and you can also ask what happens when case is not a natural number and here those work by well tomorrow navigate Brown who got the same thing Law bond when would carry is supposedly rational number but the bonds so you can ask if you have a bone for all rational was wouldn't you have it for all real numbers but the reason is that the the bond that they obtained will depend upon the height of national so the concept that the Nets Wilson here will depend upon K but it doesn't depend just on depends on the type of case so you have a small slightly weaker results for all real numbers game the military was described by this sort work with that work through it and let me illustrated by 1st giving you approved that's stuff that if I want to look at the sum of 5 l have IT to the case the 2 acts could be just 1st for all OK which are natural numbers which extended to rational and the rational it in a little bit that I can get here a long ball of Form 5 extolled Dexter King landscape on so so the main thing that reducing the proof is that so I said that if I can compute the 1st moment plus along that I be able to produce results of the summit real-estate what what I mean by that I would like to be able to compute the sum of the 1st moment I'll have kindly in the fundamentals communes and twisted by some number elements and analysts some
number which I would like to be in some small power of of fax maybe I'll go so flexible was from you can do much better but let's start with so How should I estimate these well 1st you can start with up with some kind with the relation for real function and tidy which is called the approximate conflagration but 1 can there many exact words office soliciting approximate about that except the name which says that you can essentially understand the cell function by my 2 sons which go up to roughly square root of the kind that so the sound goes up to basically square artifacts kind L an over time and there are basically 2 sons of those that would be a sign of a functionally question involved but operatic characters assigns always once like-minded just 2 copies of all of the scandal so this should not really be taxes should be like a good deal pie or something like that but of little worry about and then say you if you put that in here and exchange sounds so you get this list moment is the sum over the to act and that means some more and look to the text of the letter the you were and I will listen sir so now I would like to take this sum will work over and outside and bring the summer work over the side sorry for the right here the authority the case so I think the sum and outside and bring this summer with the inside so I would like to understand something like that of some awareness through tax From some work um D after last year so now I can view so it's not just doing this as a sum work characters tied the reciprocity I can use this as a character sum To the modulus and times and that and is fairly small Italy of sizable tax analysts fairly small size only some small power facts so it's a character somewhere these pretty large and this and that was pretty small so I can really understand characters sums of up quite easily by even something as simple as children grow service the 1st larger the moment the complicated your argument is handled this should be with me but in this case even a simple but only when only quality the argument would say that would imply that I only have to worry about the towns where this is that's the the principal character that ain't done everything else this year inheritance so you only have to look at the town's where and times out of square is the main thing and everything else gives you reminded OK so what would that be if I write LSI up that's a L 1 times and to square where L 1 a square feet then held and has to be a time times some other number and square and so you can know I understand this fully the answer should be the answer should be apart from some constants that 1 always square root of a lot under sum all 5 1 over and in some appropriate range gives me a lot of fun Route X or so that's the answer for this for this moment and you can see that we can evaluate this quite easily with a very good Maine Dominic would remain the fees for small so now we can use this to get bombs fall for all the moments as follows sir I will look at the sum all else have and I try to think of something which looks like an approximation to the critical area of tightly to the K minus 1 say case an integer and a of is going to be the temptation to something that I can think of as being an approximation of the other function but not quite so this little lacks if it were like tax that it would be really a good approximation to the other countries but I would like to choose had to be some smaller Our effects may be Exelon 100 so if I choose that the fight expanding of developed some large power that's still going to be a shot the football under that I have here and so I should be able to evaluate all of this all of this August we want evaluators so finish this calculation after lunch so we can evaluate those and then on the other hand I also get just for using holders inequality that this is bounded by have tied into the cake 1 1 which came under the sum of a hiding became 1 will as well and but I can evaluate this too because this is just a shortage of former that I have to evaluate so this can also be a OK enough goods if you compare those 2 then you get a lower bound for like a moment of the left and this moment you don't use anything in this argument cells show and you get the right vital so that they stop maybe chairman of continual this is the 1st to know this very well refused to put up with it the FAA rule yeah I think that
they don't have a good of battle this honest right but went 0 for example I think it will be a Romeo right through so we don't know yet I think it's not a good yeah I don't think
anyone has a good idea of what the Constitution something that will change of power savings of so they can to the sea of carers for these interview moments also can the killer should be narrow size square root of Father of the respirable flexible water on this conviction may or may not be true but that but there's no use this new even plausible connection factual "quotation mark version we were injured after last year