6/6 Nilsequences
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Title 
6/6 Nilsequences

Title of Series  
Number of Parts 
18

Author 

License 
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  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

Content Metadata
Subject Area  
Abstract 
Classical Fourier analysis has found many uses in additive number theory. However, while it is welladapted to some pro  blems, it is unable to handle others. For example, if one has a set A, and one wishes to know how many 3term arithmetic progressions are contained in A, then Fourier analysis is useful, but if one wishes to count 4term progressions then it is not. For this, and other, problems the more general notion of a nilsequence is required. NIlsequences are a kind of «higher order character» forming the basis of what is becoming known as «higherorder Fourier analysis». The talks will be about this theory.

00:00
Sequel
Process capability index
Prime number
Event horizon
Prime ideal
Average
Selectivity (electronic)
Hadamard, Jacques
Units of measurement
Social class
Physical system
Logarithm
Product (category theory)
Cartesian coordinate system
Bilinear form
Functional (mathematics)
Formal power series
Proof theory
General relativity
Summation
Number theory
Hadamard matrix
Computer animation
Large eddy simulation
Linearization
Arithmetic progression
Resultant
05:09
Complex (psychology)
Beta function
Multiplication sign
1 (number)
Inverse element
Mereology
Weight
Subset
Sign (mathematics)
Analogy
Cuboid
Lattice (group)
Eccentricity (mathematics)
Physical system
Social class
Product (category theory)
Moment (mathematics)
Basis (linear algebra)
Infinity
Bilinear form
Functional (mathematics)
Flow separation
Lattice (order)
Linearization
Normal (geometry)
Arithmetic progression
Point (geometry)
Standard error
Computer programming
Finitismus
Divisor
Event horizon
Theory
Number
Prime ideal
Latent heat
Lecture/Conference
Average
Wellformed formula
Operator (mathematics)
Multiplication
Expression
Counting
Volume (thermodynamics)
Set (mathematics)
Latent class model
Carry (arithmetic)
Residual (numerical analysis)
Formal power series
Number theory
Convex set
Coefficient
Local ring
17:43
Linear algebra
Complex (psychology)
Standard error
Group action
Multiplication sign
1 (number)
Process capability index
Inequality (mathematics)
Mereology
Subset
Prime ideal
Mathematics
Manysorted logic
Lecture/Conference
Term (mathematics)
Average
Operator (mathematics)
Circle
Length
Subtraction
Social class
Physical system
Forcing (mathematics)
Mathematical analysis
Set (mathematics)
Mathematical model
Variable (mathematics)
Cartesian coordinate system
Bilinear form
Functional (mathematics)
Flow separation
Proof theory
Formal power series
Summation
Number theory
Cube
Order (biology)
Normal (geometry)
Right angle
Coefficient
Arithmetic progression
32:16
Group action
Product (category theory)
State of matter
Parity (mathematics)
Multiplication sign
Prime number
Functional (mathematics)
Event horizon
Sequence
Theory
Prime ideal
Frequency
Summation
Number theory
Lecture/Conference
Average
Order (biology)
Normal (geometry)
Right angle
Arithmetic progression
Resultant
38:09
Axiom of choice
Logical constant
Group action
Multiplication sign
Range (statistics)
Parameter (computer programming)
Weight
Prime number
Casting (performing arts)
Meeting/Interview
Forest
Square number
Physical law
Physical system
Social class
Moment (mathematics)
Functional (mathematics)
Sequence
Bilinear form
Proof theory
Category of being
Linearization
Normal (geometry)
Figurate number
Arithmetic progression
Resultant
Directed graph
Point (geometry)
Standard error
Existence
Inequality (mathematics)
Rule of inference
Event horizon
Theory
Power (physics)
Prime ideal
Crosscorrelation
Lecture/Conference
Helmholtz decomposition
Average
Selectivity (electronic)
Skalarproduktraum
Addition
Multiplication
Expression
Quadratic form
Summation
Number theory
Local ring
Nearring
56:49
Point (geometry)
Classical physics
Complex (psychology)
Standard error
Randomization
Parity (mathematics)
State of matter
Confidence interval
Image resolution
Decision theory
Multiplication sign
Prime number
Power (physics)
Glatte Funktion
Explosion
Prime ideal
Causality
Lecture/Conference
Average
Analytic number theory
Skalarproduktraum
Social class
Condition number
Product (category theory)
Prime factor
Analytic set
Functional (mathematics)
Sequence
Summation
Number theory
Phase transition
Noise
1:04:37
Point (geometry)
Logical constant
12 (number)
Group action
INTEGRAL
Multiplication sign
Inequality (mathematics)
Weight
Event horizon
Food energy
Grothendieck topology
Mathematics
Root
Lecture/Conference
Wellformed formula
Average
Helmholtz decomposition
Geometric series
Square number
Integer
Analytic number theory
Skalarproduktraum
Social class
Series (mathematics)
Product (category theory)
Military base
Algebraic structure
Division (mathematics)
Variable (mathematics)
Cartesian coordinate system
Bilinear form
Functional (mathematics)
Sequence
Lfunction
Convolution
Summation
Number theory
Combinatory logic
Linearization
Coefficient
Identical particles
Nearring
1:15:25
Point (geometry)
Polynomial
System of linear equations
Parity (mathematics)
Multiplication sign
Distribution (mathematics)
1 (number)
Similarity (geometry)
Mereology
Rule of inference
Theory
Power (physics)
Prime ideal
Frequency
Goodness of fit
Population density
Crosscorrelation
Manysorted logic
Lecture/Conference
Average
Analogy
Square number
Subtraction
Physical system
Distribution (mathematics)
Spacetime
Gamma function
Physical law
Algebraic structure
Cartesian coordinate system
Sequence
Functional (mathematics)
Bilinear form
Automorphism
Convolution
Orbit
Combinatory logic
Network topology
Equation
Linearization
Normal (geometry)
Arithmetic progression
Identical particles
Resultant
00:03
she b
00:06
the wound to the head of the move to a new unit to thank him here but I found the net general manager we don't want them to think the word you and in the end came in and would installment today in this last lecture is applications of the material that been discussing primes some start by stating the results the awards a sketch some proofs of understatedly the any results about pride in fact it's convenient to introduce the formal functions and that's defined by landed which is so it's going to be it and is a powerful prime and 0 otherwise and and the reason introduces is to avoid having huge numbers of logs involved in your statement said the prime numbers it is equivalent to the statement that the average value of the full Mongol function this 1 wasn't as interesting so the applications to Prime's adults took a battered by counting how often linear forms in prime and linear forms take primaries so I'm interested in finding crime values of linear forms so I'm going to be interested in a collection of linear forms select side equals PCI 1 subside sepia collection all the linear forms of filing forms so there is a great amount and that being Devera also them up said the Deaton said I say for example EG I could have sequels for the to on the phone is defined as follows outside 1 of and is on 1 side to end it is and 1 lesson to 3 events and so on and this is said this is the familiar system the system we've been talking about quite a lot this is so he backed off automatically depressions the 1 thing I'm going to talk about is how to count the number of fulltime management progressions but this is a lot more general salute to give another example of something that is covered by this so I could take um again what was 3 and the calls to and the forms as follows the soaps I 1 is on 1 side to attend to campsites 3 this summer some constant began to minors and 1 month and 2 and that if I can count how often those 3 forms of pride I would have done is represent and the sum of 3 products so can equal P 1 class P 2 class P3 so what's called the Tundra gold out of my jacket there I could stay the binary go back problems In the same formalism of course but that's not going to be covered by my name there as you probably know anyway so what is this what is a general theory about this so this is a fair amount myself entirely and done with crucial input by myself
05:09
town Ziegler that being the inverse there and for the cows would have talked about said there is if you take any collection of linear forms suppose that this collection outside Paris finite complexity of the fund precisely the complexity in just a moment about what finite complexity means is essentially that made 2 of the forms of multiples of 1 another which means that note to the homogeneous parts several callers society I don't think so homogeneous parts just what you get by ignoring the constant our multiples of 1 another and then the conclusion is that you can kind how often those forms a price 12 I'm going to cancer research in regions that suppose that case is a subset of a box of which and a midsize convex because that's what we dated the paper where we were the things I'd but really any nice set could be decomposed into complex that's so just suppose case and I said that Europeans can't ever then ,comma if you can that the intersected with the lattice points of K and you can't how often form the crime in this wasted money using the form although function and then 1 is what we call a local global principles said asymptotic leaders can be computed as a product of certain local factors basis of infinity times the product of Prime's because some paid plus an error that's expected to be small so little over and over and over the years so I'll tell you what the local factors on just the 2nd and when this won the race in the care of this what will certainly depend on the and on TV and it also depends on the size of the coefficients of the other side so the coefficients of homogeneous parts now I'm allowing of finally forms underlying the constant so much be rather big said also depends on what I ran times the constant terms but other that uniform so for example you get this stuff decay for both of these systems that I've written up on the sport said here said beta sub infinity what less religious harmony lattice .period some candy so is the volume of case has not actually not quite the volume of case because Prime's always supposed to be positive numbers says the volume of that part of carry on which and all of the 2 sides are nonnegative there is essentially just the number of points you can and the basis of peace reflects local behavior all the primes said paid to it's Canada local analog all fat and happy all of this same average is going to be the average by the end of my Reggie Evans said what he said to the date of the same expression but with local variants of the formal function so what should a local
10:14
variants of the from will function be it should be a function with average value 1 which is somehow detecting the local behavior of the prime the modern peak so what are the prime smart people apart from pay itself which is just 1 prime money infinitely many at the time that he could distributed in residue classes other than 0 and so that makes this definition quite sensible the local formal function of X is going to equal peertopeer minus 1 if X is not visible by Pete and 0 in fact Texas is divisible by so you can think of this as a local global principle for the on linear forms so in any given case with a little bit of work you can get computer exactly what this formula gives you and I'm going to tell me the outcome in 1 case the exercise as this exercise will involve while substituting into the above other than you have to do you have to remove the the weights that that I've given to the prime from the from functions said plus removing the the local weights which is a simple matter said it's a formula for the number of Fortum program at the case and progression said the number all of Kato progression In less than that of 1 its asymptotic play and the following constant said it's 1 over twice came on 1 times the product of a P and BCP times and squared ever looked to the case and where pieces of peas given the following formula said beta sub p is 1 Over P. times here the peerless ones to the K minus 1 it appears less than on the case and if P is greater care 1 might ask a minus 1 of peak times P and the Pima 1 to the Caymans 1 so it's quite complicated actually the cost of case falls on the good work as an explicit numerical value for so that's a theremin pastry ought talk about father any questions imagine not so mostly so far these lectures I've talked about just some very specific examples for example foursome progression and I haven't told about general systems of linear forms at school so many artists say couple more words about general systems of linear forms the theory all of Gallas norms eccentric um applies in the general context within the general context that I've just described said for finally complexity systems of I finally forms and so event is a big generalization of what I call the generalized 1 moment there there there is a generalization a generalized generalize from London there so remember the generalize from London firm said that if you want to count for progressions of the example against it if you want to count Fulton progressions weighted by some functions that is enough to control the Gallas norms of these functions so let me introduce the obvious generalizations introduced and operator T so sigh and I guess it also depends on the comics body came and all of them functions as 1 of the Act he would just be basically the average value so 1 entered the let's say doesn't matter too much whether I take into the deal to add that the and then some over this ,comma its body of molasses points the comics body all those functions weighted by the appropriate linear forms so is a generalization of the multiannual phrases I considered before center for it's always a bad thing just 3 simple progressions and and forwards impressions and said the generalized generalize from none of the states that if the FIA abounded we have this is bounded by garrison on the outside care at 1 of 2 F15 is bounded by some constant
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times in any of the garrison West is something called the complexity of this particular system and there's a linear algebra recipe for competing the complexity I'll tell you what it is it's not some lottery what it is and then make some remarks about so what it is it's the smallest such that for any ally I between 1 and 2 I can politicians I can petition the forms sigh I that the the forms under them side by side peace politician Sai 1 sigh minus 1 side plus 1 of the Si In 2 s plus 1 classes With the prophesy and that's so that society is not in they are finally spot for many of them a strange sort of recipes I can explain their way for example what the complexity of Fortum progression this so I thought to so for example the capital PCI equals PCI 1 out of sight for his assistance in finding Fortum progressions then the complexity is 3 so as to and the reason the complexities to is clearly if I remove any 1 of those 4 forms I can partition the remaining forms into 3 classes just the trivial classes and then the full my room removal so it may not be enough finding in a span of any of them because there independent an but I can't do it with 2 classes because if I try to do that I'd have to have 2 forms of the class and then I can write every form into variables as something in the linear spun effects sigh exercise to convince yourself about I wrote down some other ones so the complexity Of the fairground off 3 prime system is 1 and somehow what this means said complexity 1 means that the Gallihugh to norm it is all you need and as we see the guarantee to normal is something that's just classical for analysis and indeed writing the old numbers as a sum of 3 crimes is something that was done in quite some time ago by using types of Fourier analysis and thus can be handled using for analysis traditional for analysis or more accurately the party Littlewood circle method allow Hardy Littlewood Peru should have an accent on that and from the right to words of French and my 6 lectures should at least should at least get right this year the 2 of you to use it as an I mean actually we really that's what's important is that it has any value if you wanted coefficients in Q but would send values villain in Europe finally there is a reasonable person would want 1 trillion won his case so you can be handled by for analysis horny little wooden envision brought all that was done in the 1930's and then I read them 1 more example which is the example of thousands of acute so cubes and which would be of the form so I take all of the forms and 1 plus the sum over in any of them and I where a Rangers overall subsets of 2 oddity so this is to say that the minus 1 forms and the complexity it the minus 1 set in many different terms the worst so the various different types of systems of forms that you can handle now has its own understated what the complexity is wise that complexity was a bit difficult to explain but basically the proof of this generalize from none of them is gain some applications of the kg shots and quality and it turns out that the number of cases faster qualities that you need to prove this package depends on you need to arrange the forms side by various changes of variables so that this commission shots inequalities give you the right things and and when she decided upon my working at what the correct the number of Koshy shot qualities you need as it is both than the linear algebra it comes down to just a mirage of a problem to which this is a solution I don't think I can add any more intuition to that so that's a statement of the theremin and and while it's tell you some of the ideas that Karadzic approach and I think will lose nothing by specializing in the case of force aggression so for simplicity but everything I said for definitely not simplicity let specialized too the system that we've been studying sigh people's sigh
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1 of sides for all Fortum progression and so let's again rights he for the operator that it's the left 1 of the 4 therefore the corresponding operator so I told about how to you use this operation and generalize from 1 the 2 can't athletic progressions and the basic idea is that you splits the splits the carriages the function of the set you're interested in into a structured around and part so the way I did it when looking at Summer 87 so the 1st idea would be just displayed the form functions to estimate the Lambda Lambda Lambda Lambda which is what we want you might try subtracting off the average value try rising the planned as the 1 which is its average plus Landon minus 1 and then not decompose just as we did before the Lambda Lambda Lambda Lambda S T 1 1 1 1 plus 15 error terms involving large amounts the men sales maybe this is equal to Tier 1 1 1 1 and plus order of the Gallas Usrey north of London minus 1 and then perhaps 1 could show just as we did when looking at Summer 87 there that this is small and hence the number of Fortum provisions of Prime's is dominated by the it said that there are a very large number of issues with that plan so here some problems fundamentalist several problems with its and 1 is if it went it would give the wrong answer so we know it's not and the answer it would give is just wrong and the way it's always we've not seen any of this local behavior of the formal function at times pay so where they basically there's no would not taken any accounts of irregularities of the formal function modeler at 2 3 and 5 and 2nd problem it is 1 in making this session here I applied to generalize from there but it wasn't actually valid to do that because the function planet is not the application of the generalize from momentarily I was invalid and that's because land minus 1 is not blinded is not banned by 1 of its like not bounded uniformly it's understands if I can find some actual of appropriate length may be so said that's not so great then said even if there have only been valid while McCain to show that said this use Norman Lamb was 1 a new 3 is indeed small said Howard Archer poster shows physicians that little old 1 and the 2 Habomai disposal is the investor the
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inverse ceremony for the EU3 norm was again only valid for bonded functions so many states it's time that I was only it I I didn't prevent but the priest Isis gusts where any valid for banded functions for binding functions said there are 1 3 very serious problems with the plan however that the plan can be made to work by addressing all of those issues but the first one is actually the easiest to address said .period 1 it addressed using something called the WHO and the idea here is that you take doubling to be just the product of the 1st few problems and the need to take a number of crimes it tends to infinity so there is nothing really special about Loblaw and sums any slowly growing function would work but you want to make sure that it doesn't grow too quickly and consider that while and rights blunder as an average of 4 functions we shall call Landis of WP and these are defined as follows members of WB event in London W W 5 . order 5 of W landed W and plus being so that's basically saying Let's foliage vintages into progressions modular doubling their here the high school vouchers BMW is 1 and this has the effect all of smoothing out the irregularities Marjorie small problems of Lando said these new functions blandness of WP on 0 2 not out bad irregularities irregularities modulated primes p that less than the threshold that explained why that's so what and when strict comes from let me just show you whether the idea that led to this effect apartments 3 5 and 2 is a prime time is always best to ignore it but 1st although all of the problems suppose visual order happen if I consider what happens if they consider the end for which to and plus 1 is when I said get the sequence 1 2 3 5 6 8 blind 11 and 14 and you see that that is 1 no longer consist entirely of odd numbers and in fact it's half even numbers and half of numbers a half released it would be if I continued its asymptotic half a year and half or and the reason for that is that number is equally likely a prime numbers are equally likely to be 1 lot as they are 3 1 4 crimes are equally likely to be 1 Australia on March 4 and that's a result of the valor and appropriately enough I think 1 of these guys proved the prime number theory and with the With what
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was available at the time they could easily approved the version of the prime numbers there infractions 1 1 4 4 3 1 4 so that's all said that I I don't want to continue working with Landis up WBC I'll go back to working with Lander but what should pretend that I've done this trick and so it otherwise we're going to pretend from now on that lumber itself is nicely distributed In that modular small problems so will pretend the district is being applied and the number itself it is nicely addition 2 small modular so in reality you do have to have include the W B and that really the only effect that is that throughout the paper there is a W B she that a payment so 2 and 3 are more interesting I'll deal with them in a little bit together so the point about 2 3 is that although the formal function is not blinded is bounded by something that's much easier to understand than the function itself for 2 and 3 the key observations that although the latter is not blinded it is bandaged above by another function it is .period wireless find by another function new pop by a constant sometimes "quotation mark most mostloved some other functions which we call and and this function here is much better understood then landed itself and this function New comes from this is the theory of the sales 150 here was called cellblocks sell 6 Santos tell you very briefly the key idea because I feel that everybody should see this idea at some point advantage for and the amazing observation is just take any weights landed the Select landed the From equals 1 to some threshold off and so it was always going to be a small power plant hours and some small power let this be any system of weights and consider consider the following functions somebody divides and and the necessary to offer local it's something better Lamberti squares I mean so it is just an arbitrary system of weights but I do want to have land 1 is 1 to consider function like that then I claimed amid all cynicism is a nonnegative function but furthermore if any is primed and there provided it's not tiny provided it's between off I and said "quotation mark take and is also fine and then after then is equal to 1 and that's clear and it's summit devices of a prime number the road course 82 such devices and any 1 of them is in the range but I'm allowing myself so manifestly contraband is always nonnegative and so on the major rises the characteristic function all of the problems now it may not be completely obvious to you it shouldn't be an but efforts in the easiest thing in the civil White slammed at the edges and some sensible way after the easiest thing to compete with so for example if you wanted to prove if you wanted to figure out the average about while you just some analysts refer to as many expand the square and you get a summer but the point is advised relatively small it's a fairly short something you can estimated Adjutant General Egon estimated without too much difficulty provided or is not to be so it's reasonably easy to compete with cases that arises the figures often quite easy to compete with the Forest the see small but he said that there's a choice of the land of the 4 which efforts no I mean for typical choice of land at the justify choose them randomly after all that is going to be much bigger than the prime so that major rises the primes it will also be supported in on many other numbers be big so the remarkable fact is that you can't use the land to be so this is not too much bigger than the primes said the land and the can be chosen so that while for example this summer than last week at the end of effort and is bandied about by constant depending only on CD times of the day number of crimes less than an hour and that's what sellby date so the way 1 what 1 way to make a sensible choice is just too look at the expression you're interested in we want to make the total weight of a small and it is such a quadratic form in the land of the essentially so you just minimize the quadratic form and that's how you choose your wasteland so a ridiculously short course in the sellby except that this is basically the idea and was the 1 further I should say is that if you want a major rise the from Mongol functions you can take near to be Logan and simply just log began times after that since then and it is also usually a good idea to divide through and we normalize said that the total massive 1 Senate source said the West is chosen so
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that the average value of new it's just precisely 1 I yes we should also be best more for you to do so very good yes actually could probably based small but said least 1 of them should be small and it's the argument that I so GCC the average value of nears 1 and then this will major rise from Mongaup function up to a constant Salameh event will be bounded by constant times near .period so the existence of this function New allowed you to prove to say lighted crew generalize from 1 their 1st certain welcome 4 functions like the formal functions so is not binding but it is bounded by something that you can compete with and I didn't see proof of the generalize moment there but I did remarked that it uses the kitchen shots inequality and basically you do the same shots quality but whenever you were tempted to just throw away function because it's bounded by 1 you must resist that temptation and instead make sure you include the weights and then essentially the same computation works you find yourself needing to evaluates a large number of expressions involving new but that can always be done because they've been chosen specifically so that you can compete with them all right a few remarks on March and what actually because ii Jim nobody to write down what said and I want to say a few more things they want to run out of time so what about . 3 the investor certain that you could also prove the investor mother stated namely that functions with a large youth Renong correlate with class to know sequence that's also true without the assumption that the functions of bag provided that bounded by same year so it turns out and that we can prove a variant of the investor for the cows norms 4 functions bounded by by new about by new plus 1 and bounded by a constant multiple the the average of 1 plus new so that's another function which has averaged 21 an example of this would be land minus 1 which is the function that we actually care about so the theory here does not use specific properties of the exact construction of this new coming from SUV excessive but rather a host of properties about linear forms in involving knew that can be established in that particular case but rules invalid for more general functions now we don't do this by going through the proof of the investor step by step and making sure that we only used to the fact that would guided by new but this is actually a consequence of the investor as a black box so this follows from the anniversary the Baroness functions together with decomposition results this decomposition there was actually the key idea In my 1st joint where we're Terry's Tire where we prove that the arbitrarily long progressions of primes set aside from 2004 and the decomposition of there is that slammed a minus 1 said and that actually any function bounded by constant mostly New founded by and a constant multiple of new or oneplus inning or what have you so a wide range of functions you can decompose them as abounded functions bounded by 1 plus an error that's extremely small in the Gallas surplus of function was tiny and illusory and so why would this then inflate the investor Philander minus 1 well if the EU3 norm of the lefthand side is large and because this is tiny the 3 Norma the bandit function is large so I can then apply the investor for the bandit function which then correlates with a class to sequence this don't correlate with a class to know sequence because it's tiny in Renault and hence this correlates with the class to know sequence said that's the major of argument I should write down how that goes so many give these functions names at its school this little at school G Solana minus 1 in use revenge is greater than that Delta let's say implies that the 3 normal that is greater than those referred to is promises correctly and that implies by the investor for banded functions that f correlates with no sequence the inner product of F with Ky large sum class to know sequence and then that implies that land on the minus1 correlates with the casting of sequence at the says would GE does not correlate with no sequence soviet opponents of GE with is tiny and announced by the Congress of the investor so remember said that the unless there and for the gas norms comfort comes with a Congress says so the film
56:51
itself says if you have a function with a large 3 long then it correlates with a cost to no sequence conversely if you correlate with a cost you know sequence then you have a large 3 that she does not have a large Israel so we've managed to prove the anniversary for function is not bad and that is that she never addressed 1 of the problems that I foresaw I'm with my plan questions at all you need to leave the yesterday right so this is precisely there's no contradiction yet because I've not ruled out the possibility that London minus 1 does correlate with a University it would be strange if it did because land minus 1 has to do with the primes no sequences have nothing to do with the times I but at least the created 2 so the remaining task but the remaining task that is to show precisely what my notice suggested the remaining task its shares the inner product of land minus 1 with Ky and maybe I should just clarify exactly what I mean here by the end product this is the average value less began all London and minus 1 times Clive and I would like to show that this is small Little 1 for old class to know sequences on all things complexity and the error would be allowed to depend on that phase complexity so that's the task and this involves techniques that come from the classical artist of prime number 30 others envision an initial step which is it it reduces to a similar question I mean I'm oversimplifying quite a bit with what I'm about to say but some it said the fact is something that comes up quite often it is a question about crimes a related questions about the Métis function the report reduces using fairly standard techniques based on based on the fact that the formal function is what is that the compilation of the Mobius function with the smooth with a nice analytics smooth functional state so you can always take a sum over the Mongol function and writes it as a sum of things involving the Métis function and this nobody is usually very benign it reduces the astonishing that the inner product of Mobius I would just remind you of my decision was medicine a product ,comma which is the average value all knew of and confidence is a little lower than its station here is probably the unfortunate because in analytic number theory ties ways a terrific character think probably I I'll rewrite business at some point and call it something else 71 happens to watch this video precisely this point this is not directly character or that could be no cause it's an average yet said Monday and that is it is minus 1 to the power of the number of prime factors of and parity of a number of prime factors and what we're trying to prove here is an instance of what's known as I think this terminology was introduced by Friedlander and Evonne it's but it's something that's all pervasive in Alice number theory it's an example of what's called them Mateus randomness principle which is the idea that the Métis function should just be orthogonal to everything is the sum of the resolutions of the Security spends most of I will actually and everything that I would say would be true with the function that's the naval function that if you don't include that's where free condition which is in some ways more natural depends in other ways it's less natural as in many astronomers principle is that medicines orthogonal to everything unless a sum obvious reason why it shouldn't be so it's clearly not orthogonal to itself and noise is orthogonal to the formal functions but if you take a function coming from somewhere else so so coming from an algebraic construction or construction like these noble characters this simply no reason that it should correlate with the Midas function so that's a heuristic are proving it in any given case is not always so easy there are basically 2 ways that I know of approving what randomness principle the 2 methods for proving rigorously that the inner products of Mobius with a particular function it is a little over 1 some and those methods are roughly speaking the 2 cases is that this kind of multiplicative so if it is isn't directly character or is
1:04:39
just the constant function 1 and then the techniques that you would use would be techniques involving L functions the CDC to function that is just 1 so if efforts multiplicative some vague sense if it somehow multiplicative structure then we use functions contour integration types techniques parents formula exception and that is not multiplexes effect looks to be far From multiplicative then we use a different method that's called the method of by linear forms and going tell since the 2nd method that's going to be well actually I want
1:05:52
us to show that Mobius is orthogonal to class to know sequences 1 actually has to use bases in both methods and depending on what the sequences because actually constant function 1 is itself a no sequence but if no sequence doesn't look like a constant function minutes this method that's going to be used this method of by linear forms and this is associated with various names such as having and the Grotto of literature and he's bright and warm and and others a welltraveled methadone analytic number theory on the idea it's so that is a beautiful red or interested in reading Giants proved folkbanded gaps between privacy this method is also important that it doesn't come up in that may not have approached us and said the idea is to decompose commodious function Thursday compilations so it's going to be written as a son of 1 of a few functions so functions that it's been the former s stodgy and services to restrict competition some overdue divides and half of the g of energy some of them together method to work you've got to be careful about exactly how you do that in the teacher by the Rangers that those averages supported on so there's a lot of flexibility a lot of flexibility in how this is done and sometimes this 1 basic identity that governs this entire endeavor which is called Linux identity enunciated some that Mobius near than is the summer all integers case of minus 1 K times the case divisive functions them that was called the case a proper devise a function and it's a tossup case and man is the number of ways of writing down and equals and 102 and came with would be analyzed strictly better than 1 so torso OK is itself at its invited K fall directly compilation and so there's many ways this gives a lot of flexibility and decomposing into how some of duration convolution Linux identity is a fun exercise and I think I can give you a hint that will let you do the exercises in about half a lot it relies on the fact that 1 of his each which is the directly series of the Mobius can be expanded as 1 over oneplus season wants 1 of his and that's the sum of all fun is modest 1 To the by the geometric series for so if you compare both if you compare coefficients at both sites you'll find that you come out with the Linux for this is a really natural thing I mean actually for actually doing this Linux formula is not useful the reason for that is that the sum case case quite complete could be large so what 1 would actually use is a formula of his prize which is a truncation of Linux or depending on your application and for this application we don't need to much sophistication there's an identity of form this is borne by the way I know this is the English pronunciation is pretty irregular but that's about as a regular is a gets so that when I save on I mean that forms identity is popular it's easy enough to prove find in books and that would also suffice for what I'm talking about so you think envision lobbyists into at tertiary convolutions and what you so once we have a decomposition all some more directly combinations are stodgy woman let's try and evaluates the inner product of 1 of those the average a stodgy the man Clive and Westhuyzen I'm class to know sequence so I can I can evaluate the inequality of mothers with Kai as some of these things a lesser players that I've done this decomposition in such a way that I've got nice control over the support of effigies surface simplicity let me suppose that effigy all supported near 1 near the square root of event near x 2 F of X and effects are supported wags is roughly the squares and so there's just 1 case but it's a case that would could come up so the idea is take that in a product that school or something the star is going to equal while it's going to be roughly the average value over a and B of size about squaring 10 of after they achieved the baby it would be quite that there be some weights that are often that and then by Kishi Shrontz 1 application of kosher shots that at most the average it be and be primed of while the average over a year and all of this size about squarely Of I a be kind of fun AP prize so by using Koshien and and also the fact that in Japan to buy 1 that say some simplifying of Austinite here but the reminding that is I want to illustrate what the key point is which is the evaluation of the summer I to see what they see as I prefer to the change of variables just to have a look at it so this is going to be equal I'm going to quit the
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average and less than return of Clyde D & Co primed and say Here are just substituted their aid on and on b crostini and been trying to quit the PRI has declared a Philip more suggested so what is this what expanded I'd say Kaiser I know sequences all from P of the times fly this shipping bosses as well and 5 peers the prime and Bob and that is the average what it looks like a correlation of children to know sequences but you can interpret it as 1 of secrets it's just write it as the average of analysts and good and all flights on fly all of them appeared at the end of the crime and so fixed in the pride and says the squares for fixed the anticrime and indeed primed offered this is an average although no sequence on no G Cross G the gamma munchies cross itself in which the pollinaria sequence set with automorphic function flight crossfire .period and polynomial sequence and maps to the past .period end peer decried so what it boils down to is that at the end of the day is understanding how the distribution of these polymers he opponent of sequences on Gee Cross Street the different primed are related to the distribution of pay itself on tree so more precisely what 1 does is reduced in the case where PC good distributed on G I'm done it then and that most of the sequences Rico distributed as well for that you need the whole theory of distribution no sequences that I discussed an election for so similar a similar sort of technique but instead of being based on on Linux identity after rather run a criterion of doubly I believe and cut this and very interesting work found Apple gas somber and Ziegler which establishes this is running this principle for things that quite a bit more general than no sequences in fact for the nose differences come from discrete time flies on and homogeneous spaces Dumont ,comma but they deal with UniPresident on more general Jean would come at the same generalities as runners there I think so I think it's written completely the key point is that this they prove a weak result we do they just kind of a little 1 of cancelation so what I didn't say is that we actually need to beat the little 1 by big power law but I think just little 1 as the result of those those give you need this for fixed in the price and the sure on rival I think that's about all I have time to sketch so I'm going to stop them you know yeah hundreds of that's the whole what is the use of I essentially none of so all of these lectures have been about the work that was subsequent to my work with Taiwan progressions of Prime said that work is quite a bit softer than the seat you don't need know sequences of 2 that what we use here on the top board in fact that you actually To understand that paper you wouldn't need to know I mean you need the cows norms that's the important part of it but you don't need anything about no sequences world In the case of the 1st and only 1 yeah so finale and believe do you need all of this theory in general and unless it's some higher so the rest of homogeneous systems of linear forms you needed to be an analog of some so already for the system reforms X Y X plus wife there's no analysts there could have a positive density set with no solutions to equation just the odd numbers said as soon as you I had hit upon that issue you're going to need to use more elaborate theory so this just you "quotation mark right if I even wonder whether it may be the case if you have a lower fat just every system of performs this May will even automatically implies the asymptotic somehow possible that have to be compatible this he suppose you listen to what want the people who are you I actually know that you can decompose a careful convolution even just arrange it as far as the full compilation yet in general than always be bounded by 1 time and again that was just the simplistic so they'll be bonded by some power the divisive of functions but yes you only have a deal with the what we only have a deal with the Teufel compilations I think that there are in fact some parts of Zhang's work USA more carefully this triple combination structure itself the on the move in the f that's a good question I I didn't have to come so what I think maybe when you're referring to some work of as it was not completed I think maybe what I'd like to do is understand orbit In June camera at Prime return times for example I think he'd wrestled to so I don't know but I don't I don't believe and said he did not have any athletic applications that off parallel to the ones that have been discussing this election you USA yet you can evaluate oratorical flair some also will the functional gene would come I don't know what the significance of this is In general side junior interesting question officials the Russian to rule is you will be