Terence Tao: Nilsequences and the Primes, UCLA

um so a final speaker today is Professor Terence Tao and I have gone through the whole world of nationalities professor towers back to the origin almost is an Australian mathematician working on diverse fields such as harmonic analysis partial differential equations combinatorics analytic number theory and representation theory a child prodigy tower taught taught all decks numbers and letters when he was two years old being asked where he knew numbers and letters from he said from Sesame Street at age thirteen he became the youngest winner of the gold medal at the international mathematics Olympia he received his PhD from Princeton University at age 20 and became full professor he at UCLA at age 25 his list of awards is so long that I will restrict myself to listing only words that he received within the last 12 months so among them the Fields Medal 2006 the MacArthur Fellowship the sastra manoosh on price the Ostrovsky price the gym and carol collins chair of the College of Letters and Sciences and one of the top 10 brilliant scientists of the popular science magazine and I hope I didn't forget any and it's now my great pleasure to introduce professor tau who will speak on Neil SEC sequences and the primes thank you we're Christophe and thank you very much for coming here so I'm gonna be speaking about recent work in number theory mostly focusing on what I did with Ben green at Cambridge in the last few years so what Ben and I work with is an area who okay it's not so nice an area of number theory called analytic prime number theory so it's a starting at the prime numbers two three five seven just prime numbers but but not so much like figuring out which comes a prime across are not prime but picking up things like more quantitative things like how many prime numbers are there save from 1 to N and what kind of patterns are there in the prime numbers things like that it divides in two main branches because there's sort of two things two interesting things you can do a prime numbers you can either multiply prime numbers together we can add them together and the two branches they're actually quite different in many ways it's more natural to multiply power numbers together I mean prime numbers the definition is a very model clique of definition prime numbers one which can't be factored into into smaller numbers and hence actually a lot more is known about the model ticket of structure prime numbers from the additive side but this melodic focus on the additive side too so what Ben and I do is mostly on the additive side but just for reasons of analogy I want you to describe also be more applicator side subject so I'm gonna put your number theory how does this work yeah so in Mullica theory is a theory of how to express the numbers the product of five numbers so like how many prime numbers are in the product of a given number or like if you divide a number by prime what kind of residue class do you get so yeah I'll give you example divided prime by ten and you look at remainder so the last digit for prime what is what is the last secret prime look like that's a question of multiplicative prime number theory editor prime number theory is the theory of adding times together or maybe subtracting crimes together so like how can you write a number as a summer prime no moves as the difference between umbers and closely related is that whether you can find things patterns additive patterns like arithmetic progressions in the prime numbers okay I'd see so just to give you some examples of the type of results that we know about here's some results from multiplicative prime number theory if you take a really big natural number n [Music] so whatever like one we like a number of size 1 billion and you ask what does it do like what kind of properties does it have well it depends on n but you can ask statistically like for all numbers in a certain range say between 1 billion and - Billy what tends to happen with with a large number and so here are some some classical facts for example if you look at a number and you ask how likely is it to be square-free that means that it has it's not divisible with any square four nine sixteen other than one of course every number divisible by one but to be square-free that happens basically exactly six over pi squared of the time it's an interesting fact and that says due to Euler say it's a funding to teach an undergraduate class so that's that's one fact another factor of hundred years later due to a computer how many factors a big number has and on the average it is the natural log of M so if you know that a number I have this of size 1 million you know that has roughly log 1 million ten or ten or twelve factors on the average and they're very precise versions of that you take a large number it's not very likely to fly but we know how exactly how likely is you just crime the number a large number n picked at random will be prime about one of the login of the time if you ask not how many factors you have but how many primers you have then it's not login exactly much smaller it's log log in log log in register side no this is the graph this function it does it goes to infinity as n goes infinity so the bigger your number the more prime factor is going to have but but I think Kyle Pomerance Green number theorist once said that although log log n has been proven to go to infinity as n goes infinity no one is observe this in practice so you know it's really hard to come up with a number of zeros 300 prime factors that's it doesn't happen very well ok so that's that's more applicable theory prime number theory if you go to edit a prime number theory what you start finding is that the results are fewer and in fact many of them are conjectural so all those results are proven here are a typical some results from large number theory be taken again a very large number n we know for instance that any large odd number n can be written as a sum of three primes this is a famous theme of vinaigrette off from 1937 it only works when n is large enough there is a conjecture it's called the odd Goldbach conjecture that in fact every number bigger than everyone will begin what does seven or something should be the sum of three Prime's but it's it's not known uh it's not known that all numbers only or big numbers what is big mean I think the world record right now is all numbers bigger than ten to the 1,346 all odd numbers that are that big or high sums of three primes so there's a gap that people are still trying to fill ok well so that we know a big a big number n so there's a famous conjecture between prime conjecture that we know that infinite many primes so we know that any prime infinitely often but it's also conjecture that not only can you be prime in from the offer but you can be prime and also to less than a prime so you can be prime and also next to a prime so you can part of it's called in prime and that should happen infinitely often this is coupling the oldest conjectures maybe in mathematics it's conceivable that Euclid back in 300 BC thought about this problem and it's just that I mean but this is an active problem rather than oh well click it a problem and as such we still can't solve any of the basic problems in additive feel even though we know a lot but Monica do yes I'll come to that so the current prime conjecture we we don't we don't we can't prove but we can't do something so the the closest result we have on the cream prime conjecture is I was out of Chen in 73 that while we don't know that there are infinitely many Prime's that are next to a prime we do know that they're infinitely many Prime's that are next to what's got an almost prime which is either a prime or product and two primes a number of at most two prime factors so that's the part is closest if you can be to being prime without actually knowing in your prime and so we do know that there are infinite many Prime's that are close to an almost prime there is a very important reason why this the proof this team does not extend something called a parity problem it's very hard to tell apart whether you have an even number or number of prime factors but that's another story and of course you might've heard of via the even go back conjecture which is that to every even number particularly large even number should be the sum of two primes ever even though a bigger more four and so that's an odd conjecture go back I think into 1780 something okay at one point it was even a million-dollar prize attached it's expired now but it would still be be very good to do that this is about as hard as a twin prime conjecture that see the tranform conjecture is telling you that two can be written as the difference of two primes and the go back in check is telling you any large number convergence is similar to primes and they're almost the same type of conjecture but we can still beat up them okay so that's that's that so multiplicative an additive number theory they use very different techniques so if you want to understand more applicable problems like model flying times together breaking up numbers under factors and so forth it turns out that what you really have to do is that you have to understand what not just what the prime is do but you you raise a prime to an arbitrary complex number s look at PBS and you ask what that's what does PDEs to like what what what does Pelias look like on the average how is it distributed what's what is some of the pieces together what you get this type of it turns out that knowing what what the p the powers of PBS do will tell you eventually what products of primes do and the Missis be complicated but it's ultimately due to to this low algebraic fact that the product of two primes raise the s is just the product of PS and QD s so if you understand PE s and you are saying cutely yes in principle you understand what PQ s is doing so they understand what what are the primes to any s are doing then you know what what products of primes to any s are doing and you can you do that notes for every s you can start doing some sort of contour integration or Fourier analysis and you and you can you get and this is this type this idea goes back to Euler and it's its own and it leads naturally to this this wonderful function the Riemann zeta function 0 of S which has two nice representations it's either the the sum of one of mental year Society over what Richard theory which is so we know mais integral is integral of 1 over XD s you take sum of one and only s get a very similar function in some ways and very different in other ways but because of actually because of the let me take this sum can be written as a product of a primes and you see the PPS there and so in principle the understand zeta then you understand what PPS is doing except it because there's minus 1 here you actually need to understand one of those 80s in order to understand the primes and so you need to know when this thing is 0 and that's a that's a really difficult problem that's got to do the Riemann hypothesis which no one really has much of a clue what to do with including us so we don't have anything to say about women apophysis this is multiplicative stuff so we instead work with the additive theory but I just want to put this up because it's it's very well known Swiss wingman hypothesis okay so we instead work with additive problems for example how does for example the go back question is how to what numbers are sums of primes and you're working with some surprise rather than then products of primes it turns out the oldest Riemann zeta-function stuff turns out to be well not useless but the file is useful then we'd expect it so so even if you have the Riemann hypothesis and various generations of the Riemann hypothesis we don't get things like we don't advance anything on the twin prime conjecture exactly the Riemann hypothesis would be solve it with fantastic you would solve you would get really good understanding on any model play group model cricket if a question you please but it still doesn't say much about the additive questions this is a little bit but not not not not a dramatic amount so for multiplicative problems you want to send P to a power for additive problems you want to understand a power to P so here what you do is that you look at things raised to the power P so it turns out that a good thing to do is a cube is complex not the power P so e to the 2 pi alpha P where alpha is just some real number in number theory often abbreviate e to the 2 pi we see it so often either do PI IX is called a of X so this is this this expression here we if we very clear alpha P so if we can understand what this is doing this is want me to think about this and how turn this later is you have a circle and alpha is some angle and so you can you can imagine starting at one and rotating with alpha and rooting the alpha and going around in random a circle and and you have all these points in the circle so imagine you don't think every point but you just take the prime points you take the second point third point fifth point the seventh point 11th point and so forth I give you some subset of points in a circle and knowing what what those points are doing is actually an equipment to understanding what how what what is this this number is is how this is distributed and this turns out to be really key the reason why this is important is because this expression is connected to lets you understand sounds again because of this identity that eat the other people skills expression when people skew is just the product of EFP you have a cute on the laws of algebra so if in principle you understand exactly what this expression is doing this expression is doing you understand what this is doing and then using Fourier analysis or something you can then understand what P plus Q is doing so we really want to understand this expression how its distributed and in fact so rather than the zeta function in fact what we and work with attentions focus on is something called a prime expenditure some you've some eat lis alpha P for all Prime's P up to some number N and if you understand this well enough in principle you end this expression and in principle you can understand things like some surprise and this whole approach as a name is called the hardy Littlewood Vinogradov circle method and it for example this is what Vinogradov used to prove that every large odd number system a free prime Stuckey's based on this approach here okay so now let's contrast two results one from multiplicative number theory I don't know more theory so here there's a sort of nice symmetry mr without odors up do Ashley conserving primes and atom equations and that's the resulting more applicable number theory and then there's a reason was I've been in myself on not on primes and ethnic progressions but on earth equations and primes and additive result so I just wanted to put them up in the same screen so there was I would do Ashley is that you could take an infinite arithmetic equation so that's in other words of a residue class you take some modulus cube and some number eight for example Q could be ten and a could be three and so what you're doing is taking all the numbers which when you divide by Q give you a as a remainder so all the numbers which divided by 10 give you three is remain in other words all the numbers which end in three so that's an infinite ethnic progression three 13 23 33 43 some of these numbers are prime someone not crammed like three is prime 13 is prime but 33 is not prime but there is a zoom is that anyone in this intimate ethnic progressions will contain lots lots of Prime's we're going to infinitely many Prime's unless there's an obvious reason why they don't and the obvious reason is that a and Q have a factor like if you look at all the numbers that have a last digit of 2 to 12 14 as R to 12 20 to 30 to 42 they don't Canadian don't get in one prime that's - as a common factor of 10 so they're not they're not infinitely primes heading into or 5 or 6 but there were but there are infant many primes in 3 or 1 or 7 or 9 this is the Duras theorem so I think progressions can a not surprise our result which we proved over 3 years ago now is the opposite that Prime's getting lost ethnic progressions so you take the prime numbers 2 3 5 7 11 somewhere in there there'll be a progression of then 3 3 5 insulin is a progression of than 3 that we have a progression of length 4 let's see 3 7 11 that doesn't work let's say I can't think more often of my head [Music] no ok 5 11 17 23 this progression on top therefore you can find position in five and six it already gets hard I think you've wanted progressional of length 6 already you have to start looking at three-digit numbers and in fact even by even of a computer the largest progression that we've actually found especially has mentor I think 24 but we shown that in fact somewhere in the primes you can get progressions of any length you wish not infinitely long but arbitrary like there's a progression of length 1 million is somewhere than someone else's application with 1 billion there's no progression was infinitely long that's easy to see but but there are ones that are finitely long ok so these two results are they look kind of similar but in fact they they are proven by almost disjoint methods in fact when we prove this circular a famous number theorist came up afterwards and was sort of annoyed that there was a you know they had been 200 years of work on number theory and we use almost none of it I think I think that the last result we used or today spec no we used we use one result from like 1900 which I know we don't act in that you need that so we use very different methods since then we found if you want to get better results than this we do need to use a lot of stuff like this the same type of machinery that directly use and so forth so it has there has been convergence but initially those are very very distinct so let me talk briefly I mean these are both reasonably typical to them something directly through them you can you can prove an undergraduate course maybe two or three weeks and an hour theorem point you know there was a quarter I think it takes about a quarter to present the whole thing but I can give you some some some sketches so the previous results are so qualitative results and let me go back so both these are so you say so durations are told you that it's infinite many Prime's in this progression and we say there's the arbitrary long progressions in the primes but we don't say it's not like this is only qualitative we don't say exactly how many dirty this formulation doesn't say exactly how many Prime's there are up to some in in some progression and we don't say here exactly how many progressions there are in the Prime's up to some limit and so it's only qualitative and it turns out that if you want to prove the qualitative result you have to first prove a quantitative result okay to start counting how many Prime's you have here how many progressions you have here and so forth it's sort of like if you want to you know you have this box full of sand or something you want to just say that there's lots of lots of sand in this box you can start looking at all the grains and saying oh this this loss of saniye but actually the way we actually get a handle in this is said we don't a key open the box but we just have weighed the box and we say I'll get this box is very heavy and each grain of sand has a certain weight and so therefore this little a lot of sand in here and so we the way that both of these these results work is that we sort of the way how many prank the total weight of all the primes in a progression and here is a weight the tour way to progressions in the primes and you get some lower bound and in both cases on these things and that's what gives you this results so this is where the analysis comes in once you talk about bounce up about as little bounce you're doing analysis [Music] so the way we do this okay so we need some some math impatient so we have to give the prize of weight so it turns out that we want to give every Prime a certain weight and the natural weight to give every prime turns out to be a natural logarithm so the prime to wants to have a way to vlog to prime of three wants to go into log 3 and so on and so forth it's a reason for this I'll get to it in a minute but so we formalize this we define something quotable mangoat function a lambda n so lambda n is going to be log of P if N is a power of P and zero ever else so lambda two is log 2 so 2 has weight log 2 for has weight log 2 8 has weight log 2 3 9 27 these have weight log 3 525 so they weight log 5 and everybody else has weight 0 so in fact most numbers haven't have no weight like six and third and six and 18 they sit there in a way but basically the primes and a couple other things like powers of primes have some weight there's a reason for this for making this funny function but this is this is how we tell how heavy every prime is the definition another piece of notation I will keep averaging things so if I wanted to get the average value of a function you take all the numbers 1 to N evaluate f 1n and take the average we'll just call it the expect the expected value of f from one day and this is a notation which you see a lot ok so this is some notation okay so why do we play with this function it has two nice properties like you as many nice properties but but ok so what why for my good function so the first one is that you all know the fundamental theorem of arithmetic the fundable the orthotic is the fact that every integer can be factored into primes like 50 is 2 times 5 times times 5 and what exactly 1 factorization for every for every number up to rearrangement and that fact is actually you can encode it as a very nice identity you take what if you take the log of that fact so you know so 50 is 2 times 5 times 5 so that means that log 50 is log 2 plus log 5 plus log 5 and you can formalize that more generally you take log of any number and that's the sum of all the factors of that number of this form angle function lambda D and so this is if you just chase the definition of lambda D but then you see that this is true and in fact this is in fact another way to define lambda this is the unique lambdas unique function with this property so lambda is the function that encodes the phonological movement arithmetic and in many ways it's the most fundamental object that defines the primes and this this identity once you have this you can do all kinds of things that you can you can do what's called a melon transform your melon transform this and then you can do other transforms of that and so on and so forth and and you get lots of lots of really nice so other identities coming out of this bag okay so that's one fact about this lambda and then there's another really important fact about lambda which is so lambda has takes all kinds of values so as I said it's usually 0 and then every so often is really big so if you have a big prime like say 103 lamb is quite big it's log of hundred 3 which is what 2 5 or something so it's sometimes small sometimes big but on the average it turns out to have an average value of 1 that's the great theorem so the the average value of lambda up to any large number n is basically 1 plus a little arrow which goes to 0 as n goes in there but basically this function which is very spiky on the average is 1 and this is this is a great theorem this is perhaps the most important limit in and like number theory it's got the prime number theorem it an equivalent formulation is that if you count how many Prime's are from 1 to N the number of primes 1 to N is roughly n on log n that's another way of saying like you pick up number at random from 1 to n the chance that is prime is about one more game ok so these are two basic facts about this function okay so do i shyster okay so to prove the qualitative version you want to prove some sort of quantitative version which looks a lot scarier but these are much more in many ways these these more fancy versions of their system are much more interesting they give you actual concrete information about about the primes I mean they the purpose of what mathematics in general but parameter is it's not just a proof you out of the other theorem I mean but that's also a way keeping score but you want to understand more and more facts about what you're learning in this case the prime numbers you wanted and and just knowing that there's infinitely many this or this loss of that it doesn't tell you that much but if you start getting something quantitative like exactly how many Prime's are of them from A to B and so forth that's much more information and so you just want to get as much information on crimes as possible so do--she still remember is that if you have if a is co-prime to Q then there's infinitely many Prime's that I have the modulus a mod Q so there are many ways of stating this there was a splitter no proof essentially he does agency like this but essentially proves this that if you take the average value of lambda I so yeah these are all the primes of certain weight if you take the primes but only the Prime's that are a mod Q so you restrict to the primes that a mod Q and you look at what the average value is the average value of lambda restricted to the price of a mod Q and look at the average value up to some n then he proves some lower bound this is this is bigger than some number which is positive so the average density of this function is positive which means that it's going to be positive infinitely often and that's what tells you that that is that you can apparently more communion from the often this is basically what do really did he proved a lower bound on this average somewhat later people can once the primal thing was proven they're able to make this much better and not only because I say that this is this average value was bounded below by some lower bound but they could actually give a much more precise expression as to what this thing is that this this the this average was actually equal to something very specific the average value of this thing is actually not is exactly 1 over Phi of Q 5 Q number of numbers residue classes that are co-prime to Q plus a small error and this is a this is what's called a sequel wolfish theorem and that's a very important theorem in number theory this error term is not too bad it's got is 1 over log to a power and you can make the power whatever you wish so it goes to 0 as n goes infinity but it doesn't go to see all that fast a key there's there's another problem of is this this this Big O notation that means that there's a constant here this constant once a is bigger than 1 I think it's what's good what's good ineffective we know it's finite but we can't treat what it is and this there's a physic long story attached to that which is slightly and as a consequence a lot of theorems in number theory are ineffective we we say that there exists say and a solution to some problem but we can't tell you how big it is or where it is and it has to do with the fact that we can't track down of something really simple it's a single zero but that's a whole other story but but there's a much longer hypothesis that that will get rid of that and give a much better bound rather than one of a log it's one of the square root and it's much much much better number here but that we can't prove it's called the generalize Riemann hypothesis and we cancel there ok so that's some version of dirties theorem so why do you want these more these stronger versions they give more precise information as I said for example if you want to know how many Prime's there are plastic is just three once you know see the seagull wolfertz theorem I told you we know pretty much exactly how many prime numbers there are whose philosophy is 3 it's about one quarter of all the primes and law get less than similar I can I can tell you how many primes over these last digits that 37 or something that would be one over 40 in log in and so on and so forth and a plus an error and if you have to generalize human apophysis accurately there is really really small which it is in practice I mean if you'd like take it a number if you take say all the primes less than one trillion and you you can't you just get a computer how many Prime's there are less than a trillion whose last jizz in three and you compare it with this number accurate a slight boot and it's like a tweaked version this number of it they this number this prediction and the actual count yep agreed to like like four five decimal places exact you know a really good agreement which is what the Riemann hypothesis predicts actually but that's okay so our theorem have been that the primes contain arbitrary loan progressions it also has quantitative versions and as we release through them there is a lower bound version which is what proved and then there's a sharp version which we are trying to prove so again here what we're doing that we have a number cave on account progressions of primes of earth K and so it turns out the quantitative thing to do is is to compute some sort of average of the form angled function evaluated at progressions I was just just some some formula don't worry examples like what it is and what we show basically was that this whatever this number is it's bounded below as saying positive so we have a look an entre via lower bound on this expression and once you have that it's very easy to show that this infant this you get blossom some progressions of primes look at the cake so this was out yeah for K equals one and two you want to find progressions of length 1 and 2 in the primes that's very easy and so this was done by chebyshev back in 1850 you want progressions to learn three in the primes that's already somewhat tricky that was done in 39 and what we did was the general case but we don't just want a lower bound now that gives you some information but it doesn't give you what we really want and so what is so those like what we really want is correctly know exactly how many Prime's there are in a program of length prime progressions of length K there are up to some number N and the expected value of this way to expect values expected to be equal to some some constant G sub K traditionally this G's in the German factored form for reasons I don't understand plus a small error our childhood he sub K is in the next slide and so there was a specific number which this average is supposed to be and again this was done again for K equals one and two a while back he goes three was found a corporate last year we were able to calculate how many primes progressions of them for there are and we're working media on trying to get four and higher we split up on the two halves and I've finished with my heart just like this week and Ben knows neither half but that's a story okay anyway so what is this G sub K well I'll just give you a fashio formula and not not tell you much about it so it's something called a singular series what you do is that you don't work with well it's like II kind of a little bit to do a theme stalk you you don't work at the edges because these are hard the unseen carnivals in this is hard but you can work with finite models of integers Z mod p 0 to Z mod 3 Z mod 5 and these we understand very well and so you can sort of work in if you like in the Profi night version of the images you you work more two or more three more five and there we sort of know what what should happen if you work more to all the almost all primes are odd so almost supposed to be one more to if you work more three all Prime's to be one or two more three zero models with one exception and so on and so forth so the primes mod P should be distributed in all the ways do classes except 0 mod p and so this and so you can you can work out a sort of an analogue this moment function mod p and so every mod P gives you what's got a local factor and then what you do is you just take all those product local factors and more time together and this gives you your guess as to what is going on to image it so the some of the philosophies of imagery behaving like if you like the profile integer so all you see my piece put together so that's this this was all conjectured by Hyde a little bit eighty years ago and backed up by a lot of numerical evidence so there's this thing called the Hydra prime to post conjecture which predicts how many patterns of almost any sort should exist in the prime so wonderfully think about number theory ask opposed to other fills mathematics is that we can't prove everything we want weird but we can conjecture things that we want really well and that so almost any question other primes we can we can spray completely whether true or false whether something's true about the primes and we have lots of numerical evidence lots of heuristic evidence but we can't prove most of that ability conjecture somehow about 100 years the head of ability IQ prove things so you know I mean well we have always conjectures and will provide 99% certain they will true which is we can't prove any of them which is well it's not that bad a situation have it's better than having no conjectures at all but we'd like a little progress anyway so there is this conjecture to the prime to post conjecture which basically tells you how often it predicts how often a certain pattern prime should appear and it is very general it predicts for example how many twin primes there are up to some number n and it predicts this infant many it predicts how many ways you can you can write in even numbers summer to primes and it will predict the go back inject it predicts all kinds of things and we would really love to prove this conjecture and we haven't done so yet ok so these are these numbers just to give you some flavor of what these numbers are so these are much too so G 1 is 1 G 2 1 G 3 is 1.3 g4s 2.8 so there there's computable numbers and these are sort of this tells you sort of the average weight of all the progressions room for the primes average weight of all progression and 3 in the primes yeah so for example one that we have is that if you want to count how many progressions of primes there are other than for less than n turns out we know pretty much exactly how many Prime's there are now of how many of these patterns are it's about 0.476 in split of a log for it and so we have these very quantitative information and there's lots of other result this is just a sample of what we can do now okay so going back to multiplicative prime number theory the way we prove things in multiplicative prime number theory like like doosh a system is we rely a lot on muscle mass of identities so that lots of formulas and we have very deep and somewhere deep somewhere someone also deep and connected to to all kinds of lovely areas of algebraic number theory or Galois theory so forth again I keep again connecting two to their face talking stalk but so for example to to prove do a face through them that that that every FAS cook after Cubans infinitely Prime's more Q essentially if you understand each of these if you understand what these four equations mean then you can pretty can prove to trace through them basically you're done but I'm not going to prove to our 16 here just wanted to do to show you or just point out that you have four identities and you can unsigned these four identities then this fiscal all you need so this identity this is this is the model ticket of for you in version 4 so you finally count primes which are a mod Q you can write that as what's called a character Sam so then you Facebook these funny things lambda and Cayenne so you mess around of this a little bit and you sum it and whatever and it turns out this thing is equal to something very nice it's it's equal to L prime minus L prime over L where L is input L function strangely enough I won't we well that is but once you this identity you can apply complex analysis and you can solve this you can solve solve for this in terms of in terms of this and there's a way to do that by cauchy's theorem and what it tells you roughly speaking is that this expression here which is which is what you need to count primes in progressions is equal in some sense to a very explicit former which is actually the sum of stuff although the zeroes of the cell function plus some other things which I want to talk about why is this equals in quotes well it's sort of a cue the same problems in Richards talk this sum does Nike converge but we have ways to renormalize it and massage this and working distributions so what this actually can be made made sense if you are willing to suspend enough disbelief so this tells you this formula tells you what the primes are doing as long as you know what the zeroes of this l function are and it turns out that all the zeroes are easy to deal with except as you or one when we always won then this thing cancels and that turns out to be a really big problem so you need l1 Kai be nonzero and then what comes in is this amazing formula for the class number four which expresses l1 Chi in terms of a bunch of algebraic stuff which comes from a queue the gamma extension of the images using this cap Takai and okay so there's all kinds of stuff in here but the basic point is that is that none of these numbers are zero and so whatever this formula is it tells you that this is nonzero and so one is not a zero and so actually once you have these four facts you get do a chase there so it's an amazing an amazing proof but just want to do illustrate to that yeah okay but it's very algebraic so what we do we sort of use almost the opposite philosophy somehow you know most of the progress in a number theory in recent years has been by pushing this algebraically pointed really really hard and it had some really good results you know Fermat's Last Theorem was proven by by I really sort of hardcore algebraic number theory which is connected in some ways to do what I discussed and we lost some other progress coming out of that too but once you go to edit a prime notes our prime is really don't want to be added at least so once once they do you lose all the algebra and you have to just use analysis and okay I mean it's good for me because I don't algebra I do analysis so so I'm much more comfortable with this type of number theory so other preneur theories is much more analytical much more sort of statistical you don't ask for magic identities of other primes because they tend not to exist when you look at a data structure instead you you ask for well statistical properties of the primes not what the prime is exactly equal to but sort of how they be so how they correlate with things how they discs are lit with things especially auditory structured things so example a good example is you take the primes and you ask as I said before how do the primes interact with say this complex exponentials a linear phase e to a linear function here and you can ask how do the primes do the primes do anything interesting at all with respect to this this function now how do you define interesting well in many ways when you one thing you can do is that you just sum the primes of this way usually alpha and this is very much like this prime expansion of some I talked about earlier and so you can even ask for the four you can ask you can correlate the Prime's with this this funny function yes is there any college in order do the primes look like this function orderly not look at this function and so sometimes they do sometimes they don't depends on what is alpha is if alpha is rational it was a over Q then this is a nonzero expression it's what's called a managing some and what there's a formula but never mind the formulae to some some some very algebraic understand expression but if you're if you're trying to call it lambda some irrational you get zero so this is this is a very typical fact and so this is in fact this this particular fact is very important in the theory well versions of a more quantitative versions of this but but what you see is that so what this telling you is is that sometimes the Prime's behave like like like Ethiopia so but the parts of some connection with Exponential's when your phase is rational and not when it's irrational and so you see this this economy and this the economy seems to be the tip of the iceberg of of something a much bigger that we've so discovered in the last few years so it comes from so it's what we've what seems to be the case is that if you want to understand more ticket number theory you have to learn algebra okay you have to learn number of exemptions Galois Theory also really so high-tech stuff we want to learn to something additive number theory you have to learn a different type of mathematics and in particularly under standard dynamical systems or a Ghatak theory the is it's how to describe dynamical systems is just well just the evolution of any system which anything which moves around studying anything of a time variable basically and so in particular what I just talked about with these phases are you can you can recast it seems that what you should be doing is you should be understanding it in terms of of what's called the circle shift map on the circle so the unit circle circle you can look at the shift map by rotation by alpha so you pick some angle alpha and you just rotate by alpha and so that gives you a point in rotate and rotate and rotate and rotate because you whole bunch of points and depending what your angle is this rotation does different things example of alphas rationale so suppose you take a circle and just decide to rotate the circle by a quarter rotation like 90 degrees again and again and again if you do that that's a very boring dynamical system but you take a point you rotate 90 degrees 90 degrees 90 degrees all you do is you just get full points over and over again okay you don't get to visit anybody else in the circle you just get full points open over again and so we say that that's this flow is a periodic flow if instead you rotate by an irrational angle for example if you rotate not by a quarter rotation to say one two pi of a rotation so that's one Radian is about this much you rotate like this rotate one Radian true radius the radius four radians you know because one of the two PI's are rational you never go back to where you start and in fact what happens is that you fill out the entire circle well that's not quite true you you flower still a dense subset of a circle you get arbitrarily close to any point in a circle and and okay and that because of that we say that this flow is ergodic like it's a bit better let's go to the ergodic as a technical technical thing so you can see that that sometime because is so different dynamic assistance of difference of personalities some are periodic and some are the opposite of periodic is ergodic and the primes the primes sort of like the periodic systems but they want to have nothing to do the totally forgot existence what that means it's like if you go to the quarter rotation where you go rotate by one-quarter rotation and you take a point and but you don't look at every orbit you look at this that you look at the prime mover to look at the second point third point skip the fourth Michael a fifth point seventh point eleven eleven point 13th points of Eva you want to take the points you only take the prime numbered points of the orbit you don't get to visit all four points in fact you only could visit too well I mean because the primes are odd actually one exception and because of that you don't visit all four points of your orbit you only visit two of them and so they when you look at what the primes it how the primes sort of interact with with this quarter rotation system they have some detectable bias they don't like they don't like two of the points that they want to cluster only the other two four four points and so there's no bias when you call it Prime's with a rational rotation but and so some are rational rotations become very important in understanding this episode subject but if you instead look at irrational rotations it turns out that there is there is no substance there's no such conspiracy that the primes don't care they have nothing they want nothing to do with a irrational rotation if you rotate by one Radian no time when reading to radiance the radiance for radians so that would distribute itself uniformly on the circle if you look at the prime point second point third point five point seven it turns out this ones will also distribute uniformly on the circle is this of no distinction and so the primes they're basically just don't care about system and so this also turns out to be important so there's sort of this reason you have understood now how to approach problems in prime number theory in additive prime number theory which is basically to work out what - what dynamical systems the primes like whether or whether they that what they call it with and what I don't cry to it and once you figure out all the information you can work out just from that how they interact with various dynamical systems you can then start counting how many patterns of a certain type there on the primes so it's some very very vague philosophy here so the primes went to end we know how many Prime's are out there I've density about one of a loggia so that there you take it almost one to end the primes inside of just some sort of random or some some set inside here and we know how big the set is but we don't know how the Prime's are distributed maybe they example the current prime conjecture maybe they're they like to be close to each other maybe primus want to be next to some distance to a pathway each other but maybe they actually want to repel each other they don't want to be close to each other maybe they look after while there are no turn primes we don't know so we don't really know what this set is doing but the philosophy that's emerging is that sort of by default I think what the primes want to do is it's hard to talk or see you can't assign no element objects but this is useful for for thing about Facebook so the problem well want to behave randomly okay they would they want to give every number an equal chance being prime if they actually were random then you could you could do all kinds of things like if you had a you had a random set of 1 to n of density one of log 8 it's really easy to count how many twins there are like if you knew the power to speed randomly it's not hard to argue that the number of prior twin Prime's say from when n should be that should be roughly this number so they sort of want to be disputed evenly but they're not and these sort of that they're constrained they have some patents or give like conspiracies they they have arranged to to organize themselves in certain ways for example primes almost all crimes are odd ok so because of this so they basically hate it so all even numbers are basically excluded from for membership in the primes this is why for example you can never find any adjacent Prime's P P plus 1 other than two and three although on the other hand since since you was true since you're squishing the primes into the odd numbers that should increase the chance that your increase in amount of print rhymes because it appears odd than people's to show us be odd and so somehow this conspiracy should reduce some statistics but increase others okay so we do know that there are some sort of patterns that crimes have they're all odd with one exception they're adjacent to a mobile six with two exceptions and so on and so forth and so it seems like the way to approach these subproblems is to try to figure out all the patterns of the primes have like what what do the Prime's do I do actually so list this all the possible correlations that are have various things what all collisions that could actually affect what your trick that you're working with work out all these things are and once you have all that you should be able in principle to compute your original problem for example computing well we have your computer in Prime's but we can compute other things now can compute things like how many progressions of a certain length there are in the primes and well how many ways you can write in numbers we can for some reason we can't write fix we can't compute how you can write numbers sum or two primes but we can compute how we can write numbers of sum of three Prime's there's a good reason for that which I'll into here but there are many statistics that you can work out once you know sort of what the primes what conspiracies that the primes will will exhibit which ones that they have nothing to do with which ones that they don't and so basically we want to know how the primes interact with dynamical systems and that seems to be the way to move forward in this business we have as I said we know how to conjecture all kinds of things we believe that the primes the only conspiracies that the primes actually have should be the ones that accurate are so obvious the one thing like that come from simple algebra so as I said you know the primes all or almost all oddness obviously comes from algebra probably almost all collaborative just to three so for that most obvious and show me and the crowds were positive like it is another obvious fact that's important but apart with some obvious things there's we know most most special conspiracies in the primes that we don't know about there was a book by Carl Sagan contact which got me to a movie starring Jodie Foster in the movie what happened was a Geordie Foster aliens she went off and visited the aliens and that she came back but she had no physical proof that she a key visit the aliens and so no one believed her when she came back in the book actually the stories were different so they the protagonist goes to outer space or whatever talks to aliens comes back and again she has no physical evidence of them of of her visit but she does have proof that you did she contact alien telogen sand the proof was that the aliens told her that that they had found a secret message encoded in that if you expand pi to a trillion digits and you look at the trillion thoughts after to the point there eventually I'm so the digits will be random then eventually the way this lovely pattern inside the prime search which you can arrange into some soap it cures which they believed a message from the creator of the universe and so and this was how she could prove that she had contacted aliens because she came back and she then set up a computer to check this and they found the pattern that was the end of the book store that never made it into the movie somehow math never somehow ever gets cut out and the movies gifts but anyway you know it could be that the primes you know they look kind of random and then and then eventually they or they all start for example favoring the digit 3 or something or prime start having lots of threes in them right that shouldn't happen but we can actually maybe that one we can prove but ok but there are lots of patterns which which which might be true we do a quick recap every I said it could be that the primes after a while they just don't want be close to each other the primes never want to be within two distance to of each other after a certain point we can't stop that from happening we don't expect it to happen that we very weird but we don't we can't prove it yet but we believe that the primes have no other patterns other than those that we already know of and based on that belief which we can continue our based on I believe we can we can predict all kinds of things for the prize we can predict almost anything all other primes if we believe that they are random up to all the patterns that we already know about and that would predict all kinds of things the Riemann hypothesis twin prime conjecture with weak this heuristic predicts this is the source of our predictions and it has some very impressive numerical evidence and also now recently some some theoretical evidence as well but it's still I mean just this is not even a conjectures it's just a belief in really no way of IQ proven was there let's see which way you want to talk about this you did you to do yeah skip skip a lot of this well let me okay let me just say a couple of things so it turns out that the more complicated the pattern you want to to count in the primes the harder you have to work because it turns out there are more and more different types of conspiracies the more and more dynamical systems that could conceivably jump in and distort the count of how many of how many of these patterns are some of this is more more stakeholders somehow in the whole business if you want to cut progression of than 3 then yeah then you always have to deal this is good code what I call called linear conspiracies like if you want to count how many times I've had impose iron goes to our appears there is linear conspiracies you have to deal with and so what happens so never mind the primes it's more general question if you want to count a function of n function n plus R of n plus 2 R you know F some average value which echo F a GSM average value or eg and hsm average value eh all things being equal if you expect these these things who so independent this product should be sort of like the average of F times the average age was after age this is what you expect it wasn't there's nothing funny going on so any this this funny expression should so split into these these these these factors but as you know you you can't actually interchange products in some you know someone a freshman algebra so much easier if you could but you can't let that that is false that this this expression F of n G of n plus R hmm SR is not all wait doesn't always factor and the reason is because of these these funny conspiracies that if you said f to be a certain linear phase function G to be another linear phase function and it should be another linear phase function there's a certain identity that connects n n plus RN plus Q R and what it does is makes all these phases cancel and it makes this this average nonzero even though these functions of average zero never mind where it comes from but well it's related to a flowing very very simple fact if you have a function function of a line and you know what what is value is at one point N and neljä point n plus R if you know what if you can you know two points two points determine a line and then Third Point you know actually what what whatever you can extrapolate and so this gives you a sense of constraint between behavior at one at three different points of an effect regression and every time you have a constraint that distorts that's like that's a possible conspiracy that can distort your sistex so you always have to deal with these these linear Exponential's and so yeah there's a whole theory of this is called the circle method which I guess I will not talk about a long time so I'll skip all that blah blah blah blah blah okay once you go to higher progressions like progressions about four it turns out that you not only have to accommodate all this linear conspiracies within disease is these other conspiracies that sort of climb on board and what you want to be represented to so if you want to come progressive them for then there's there's a there's a quadratic conspiracy that comes in as supposed to get to type of it this is really the fact that if you take a quadratic polynomial a parabola if you know the value of parabola at two points that doesn't tell you what happens everywhere else but once you know the value of probably at three points or progression you can extrapolate and get the fourth the garage interpolation and so for good so quadratic functions induce a constraint between progressional than four and it's because of this fact that you really have to deal with with quadratic functions as well you have to understand what the primes are doing the structure group quadratic functions during example question four and that's that's very annoying and in fact was even worse than that there's some let's skip this it's not just quadratic things it turns out that that deep you can classify all the dynamical systems that cause trouble for progression therefore and that this turns out to be something good okay some go to two-step no system which are it's like it has to do to 700 ordinary groups which may be asked about is completely skipped it's not it's not elementary and then if you wanted to do quicker than five you to do a three step Newport in groups and you just get worse and worse and worse but we have at least as very recently been green and I understood this these twos double important no manifolds and and how they call it arrives this took three papers each of our sixty pages but we managed to do it and as a result we can do things like count progressing than four and that is that is we're at right now and we're working now on pre-christmas five six and so forth and other patterns of this type I wanted well okay there is a connection to a really nice feel good ratna's theorem but I think I don't have time for this and it's really quite technical so maybe I think of a stop here so thank you very much [Music] at one point you have a relationship where yes again rational alphas give you it's ultimately yeah there's some so constable of a rational flow basically if the orbit is finite then you get the amande then you get a strong correlation and the orbit is is spreads out over an infinite and infinite set then the coalition is zero that's basically what happens so this valency which I'm not talking about describes what what orbits of on a dynamical system look like and they they they either look like finite finite sets or or the distribute inside inside some some bigger high dimensional object as it turns out yes so this this is a so I mentioned earlier that the the only conspiracies that expect the primes to obey us are the obvious ones and this is a very concrete manifestation was fact that that we know the the primes are strongly biased towards periodic systems but in almost any other any longer existent the primes have no no correlation with and them that and this is what analyzable all these results well they were okay well it's a very class a very classical subjects when the first first things that I mean ancient Greek studied them so there's a lot of classical interest in them a Hardy was was once very proud of so he worked in a number theory and he was he was in fact yeah he was he was boasting in fact that that that he was working in the best type of medics that the purest form ethics and ethics were had no application whatsoever he's very happy about this in somebody you don't retain today these these applications you know and then several decades later people find out that these prime numbers are very very useful cryptography and so you know the your your your bank transactions open Internet or through ATM machines so forth it they're all governed by prime numbers and em and in the belief that certain operations connected the prime numbers do behave randomly in a certain sense if there was secret patterns in the prime that that you didn't know about but I did could potentially a crack whatever communication ever your bank using these patterns and you know and so it's good to know that there you know we want to know that the primes haven't have no special patterns in them what what I do here is not directly related to cryptography but but there are well there are some some weak connections though there are certain cryptographic schemes which have recently been proven to be at least weekly secure based on stuff which is regularly to do list it there are some some some infections but well I'm a pure mathematician you know it's a I work on things because they seem to produce fruit and weather well they produce more pure mathematics and then hopefully eventually will connect to something applied which happens I can more often than then satisfies me often but it's not something that you sort of force you just sort of explore what's what's productive and often you get a nice surprise maybe I can comment on the RT story so the reason re was proud of it that is math and altercations that at least my math cannot be used to make any wars was a time where the nuclear bombs were invented and so on and of course cryptography is now used yes then there is there is a lot of work on on numerical number theory I mean so the Riemann hypothesis has to do with zero is the zeta function and people have computed trillions of zeroes of stated function and I kill a lot of really Alexis came about initially formally Mirko evidence things like gap spacings of all these zeros and so forth computers in this business they're they're not as well they do some impressive results but the thing is you know the best computer today can count can basically count prime say up to about 10 to 15 or 10 to 18 if you really push it but a lot of phenomena we actually care about you can only start manifesting itself and much much bigger scales like 10 to 100 or something we just well beyond what mod Committee can do I think there's some famous conjecture of Littlewood about the demurrer primes less than X was always thought to be I think he less than a big thing bigger than something good to put a log at the gonna go of X and had very impressive numerical evidence that that's that this was the case but eventually they proved that that that this conjecture was forced eventually but I think eventually was like like 10 to 1000 or something so it sometimes número even if it's not only it can be misleading but people do use it yes yes so my thing with Ben is that the the primes contain arbitrary along ethnic progressions so I had a later paper saying they take the Gaussian Prime's he's a become so the Gaussian images are like ordinary jizz but with a complex that was like 3 plus 5i is a it's a Gaussian integer because the notion of Gaussian prime is the gas integer that can't be factored into smaller gas in it so these these forms like sort of like stars and kidding was like stars in the sky so again another movie connection there was a so there's a movie A Beautiful Mind where at one point John Nash wants to impress his girlfriend so he asked each he asks his girlfriend would name any shape any shape at all and she says an umbrella I think and so he looks at the night sky he looks a little bit ok I see an umbrella and he picks out like 12 points which which form an umbrella and she's very impressed and so you can show that this that this this about this is this is true for the Gaussian price we think the Gaussian crimes as a whole bunch of stars given any shape like an umbrella like irrational coordinates but okay in any shape you can find somewhere in the constellation Gaussian crimes in a shape which is exactly the proportions of all this under Sunbrella so yeah so the gas sometimes contain constellations of any shape between yes right yeah so so so too often demand special treatment in this business yeah and so one thing we often do is that we take the primes we subtract one you divide by two and that's sort of so the well so the primes are all odd except for two E's but if you subtract 1 and divide by 2 then you get a new secrets and and this sequence is both even and odd and what you've done is you so I projected out - so - normal exists somehow as in a certain sense once you do that and every time you actually know of a pattern the primes it's a way to sort of quotient it out and so and you can sort of rewrite your problem to ignore it it's it's a standard 1/4 bunch all set this is no it's not the known unknowns or the department's unknown unknowns it's there it's the pattern that we don't know whether they exist or not which caused the problems but the other things like being odd co-member 3 we understand those very well we know how to do them point out we have a closing reception I think this is wonderful day [Music]

Channel: UCLA

Views: 110,831

Rating: 4.9261537 out of 5

Keywords: ucla, uclachannel, professor, terence, tao, nilsequences, primes, prime, numbers, fields, medal, colloqium, science, mathematics, math

Id: XpocOKj0lxs

Length: 65min 3sec (3903 seconds)

Published: Fri Jan 30 2009