This week I am in giving this years Simons lectures at MIT together with David Donoho. (These lectures, incidentally, are endowed by Jim Simons, who was mentioned in some earlier discussion here.) While preparing these lectures, it occurred to me that I may as well post my lecture notes on this blog, since this medium is essentially just an asynchronous version of a traditional lecture series, and prime numbers hypertext capability is in some ways more convenient and informal than, say, I am giving three lectures, each expounding on some aspects prime numbers the theme the dichotomy between structure and randomness, which I also spoke about (and wrote about) for the ICM last August. This theme seems to pervade many of the areas of mathematics that I work in, and my lectures aim to explore how this theme manifests itself in several of these. In this, the first lecture, I describe the dichotomy as it appears in Fourier analysis and in number theory. (In the second, I discuss the dichotomy in ergodic theory and graph theory, while in the third, I discuss PDE.) The dichotomy between structure and randomness to apply in circumstances in which one is considering a high-dimensional class of objects (e.g. sets prime numbers integers, functions on a space, dynamical systems, graphs, solutions to PDE, etc.). For sake of concreteness, let us focus today sets of integers (later lectures will focus on other classes of objects). There are many types of objects in these classes, however one can broadly classify them into three categories: A recurring question in many areas of analysis is the following: given a specific object (such as the prime numbers), can one determine precisely what the structured components are within the object, and how pseudorandom the remaining components of the object are? One reason for asking this question is that it helps one compute various statistics (averages, sums, integrals, correlations, norms, etc.) of the object being prime numbers For instance, one can ask for how many twin pairs {n, n+2}, with n between 1 and N, one can find within a given set. In the structured set A given above, the answer is roughly N/2. For the random set B given above, the answer is roughly N/4; thus one sees that while A and B exactly the same density (namely, 1/2), their statistics are rather different due to the fact that one is structured and one is random. As for the prime numbers, nobody knows what the answer is (though it is conjectured to be roughly The problem of determining exactly what the structured and components are of any given object is still largely intractable. However, what we have learnt in many cases is that we can at least show that an arbitrary object can be decomposed into some structured component and some pseudorandom component. Also there is often an orthogonality property (or dichotomy): if an object is orthogonal (or has small correlation with) structured objects, then it is prime numbers pseudorandom, and vice versa. Finally, we are sometimes lucky enough to be able to classify all the structured objects which are relevant for any given problem (e.g. computing a particular statistic). In such cases, one merely needs (in principle) to compute how the given object correlates with each member in ones list of structured objects in order to determine what the desired statistic is. This is often simpler (though still non-trivial) than computing the statistic directly. To illustrate these general principles, let us focus now on a specific area in analytic number theory, prime numbers that of finding additive patterns in the prime numbers {2, 3, 5, 7, &}. Despite centuries of progress on these problems, many questions are still unsolved, for instance: On the other hand, we do have some positive results: As a general rule, it appears that it prime numbers feasible (after non-trivial effort) to find patterns in the primes involving two or more degrees of freedom (as described by the parameters n, n in above examples), but we still do not have the proper technology for finding patterns in the primes involving only one degree of freedom n. (This is of course an oversimplification; for instance, the pattern n, n+2, n, n+2 has two degrees of freedom, but finding infinitely many of these patterns in the primes is equivalent to the twin prime conjecture. If however one makes a assumption, one can make the above claim more precise.) One useful tool for establishing some (but not all) of prime numbers above positive results is Fourier analysis (which in this context also known as the Hardy-Littlewood circle method). Rather than give the textbook presentation of that method here, let us try to motivate why Fourier analysis is an essential feature of many of these problems from the perspective of the dichotomy between structure and randomness, and in particular viewing structure as an obstruction to computing statistics which needs to be understood before the statistic can be accurately computed. To treat many of the above questions concerning the primes in a unified manner, let us consider the following general setting. We consider k affine-linear forms For instance, the twin prime conjecture is the case when k=2, r=1, Because of the obvious structures in the primes, the answer to the above question can be no. For instance, since all but one of the primes are odd, we know that there not infinitely many patterns of the form n, n+1 in the primes, because it is not possible for n, n+1 to both be odd. More generally, given any prime q, we know that all but one of the primes is coprime to q. Hence, if it is not possible for Another obstruction comes from the trivial observation that the primes are all positive. Hence, if it is not possible for It is conjectured that these local obstructions are the only obstructions to solvability of the above question. More precisely, we have This conjecture would imply the twin prime and Sophie Germain conjectures, as well as the Green-Tao theorem; it also implies the Hardy-Littlewood prime tuples conjecture as a special case. There is a quantitative version of this conjecture which predicts a precise count as to how many solutions there are in a given range, and which would then also imply Vinogradovs theorem, as well as Goldbachs conjecture (for sufficiently large N); see this paper for further discussion. As one can imagine, this conjecture is still largely unsolved, however there are many important special cases that have now been established - several of which via the Hardy-Littlewood circle method. One can view Dicksons conjecture as an impossibility statement: that it is impossible to find any other obstructions to solvability for linear patterns in the primes than the obvious local obstructions at primes q and at infinity. (It is also a good example of a local-to-global principle, that local solvability implies global solvability.) Impossibility statements have always been very difficult to prove - one has to locate all possible obstructions to solvability, and eliminate each one of them in turn. In particular, one has to exclude various exotic conspiracies between the primes to behave in an unusually structured manner that somehow manages to always avoid all the patterns that one is seeking within the primes. How can one disprove a conspiracy? To give an example of what such a conspiracy might look like, consider the twin prime conjecture, that of finding infinitely many pairs n, n+2 which are both prime. This pattern encounters no obstructions at primes q or at infinity and so Dicksons conjecture predicts that there should be infinitely many such patterns. In particular, there are no obstructions at 3 because prime numbers can equal 1 or 2 mod 3, and it is possible to find pairs n, n+2 which also have this property. But suppose that it transpired that all but finitely many of the primes ended up being 2 mod 3. From looking at tables primes this seems to be unlikely, but it is not immediately obvious how to disprove it; it could well be that once one numbers say, To look for other conspiracies that one prime numbers to eliminate, let us rewrite the conspiracy all but finitely many of the primes are 2 mod 33 in the more convoluted format where prime numbers is the fractional part of x. This type of conspiracy can now be generalised, for instance consider the statement Again, such a conspiracy seems very unlikely - one would expect these fractional parts to be uniformly distributed between 0 and 1, rather than concentrate all in the interval [0, 0.01] - but it is hard to rule this conspiracy out a priori. And if this conspiracy (*) was in fact true, then the twin prime conjecture would be false, as can be quickly seen by considering the identity which forbids the two fractional parts on the left-hand side to simultaneously fall in the interval [0, 0.01]. Thus, in order to solve the twin prime conjecture, one must rule out (*). Fortunately, it has been known since the work of Vinogradov that and we now see the appearance of Fourier analysis in this subject. One can rather easily concoct endless stream of further conspiracies, each of which could contradict the twin prime conjecture; this is one of the reasons why this conjecture is considered so difficult. Let us thus leave this conjecture for now and consider some two-parameter problems. Consider for instance the problem of finding infinitely many patterns of the form n, n+n, n+2n+2 (i.e. arithmetic progressions of length 3, but with the last element shifted by 2). Once again, the prime numbers (*), if true, would obstruct solvability for this pattern, due to the easily verified identity which is related to the fact that the function More generally, we can see that if the primes correlate in some unusual way with a prime numbers character for all positive integers n; this identity is a restatement of the fundamental theorem of arithmetic, and in fact defines the von Mangoldt function uniquely. The problem of counting patterns {n, n+n, n+2n+2} is then roughly equivalent to the task of computing sums such as where prime numbers shall be intentionally vague as to what range the variables n, n will be summed over. We have the Fourier inversion formula where is a sum very similar in nature to the sums Thus, if one gets good enough control on the Fourier coefficients I have glossed over the question of how one actually computes Fourier coefficients where and then one uses the irrationality of Now suppose we look at progressions of length 4: n, n+n, n+2n, n+3n. As with progressions of length 3, linear or Fourier conspiracies such as (*) will bias or distort the prime numbers count of such progressions in the primes less than a given number N. But, prime numbers contrast to the length 3 case where these are the only conspiracies that actually influence things, for length 4 progressions there are now quadratic conspiracies which can cause trouble. Consider for instance the conspiracy This conspiracy, which can exist even when all linear conspiracies are eliminated, will significantly bias the number of progressions of length 4, due to the identity which is related to the fact that the function where [] is the greatest integer function. The point here is that the function The method also works for other linear patterns of comparable complexity to progressions of length 4. We are currently working on the problem of longer progressions, in which cubic and higher order obstructions appear (which should be modeled by 3-step and higher nilsequences); some work related to this should appear here shortly. |
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