Equidistribution and Primes

We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine n-space and indicate what can be established there. The methods used to develop a combinational sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic combinatorics. Applications to classical problems such as the divisibility of the areas of Pythagorean triangles and of the curvatures of the circles in an integral Apollonian packing, are given. ∗This is an expanded version of the lecture that I had intended to give at the conference honoring Bourguignon on the occasion of his 60 birthday. Equidistribution and Primes 2 (1) I have chosen to talk on this topic because I believe it has a wide appeal and also there have been some interesting developments in recent years on some of these classical problems. The questions that we discuss are generalizations of the twin prime conjecture; that there are infinitely many primes p such that p + 2 is also a prime. I am not sure who first asked this question but it is ancient and it is a question that occurs to anyone who looks, even superficially, at a list of the first few primes. Like Fermat’s Last Theorem there appears to be nothing fundamental about this problem. We ask it simply out of curiosity. On the other hand the techniques, theories and generalizations that have been developed in order to understand such problems are perhaps more fundamental. (2) Dirichlet’s Theorem: In many ways this theorem is still the center piece of the subject. Like many landmark papers in mathematics, Dirichlet’s paper proving the theorem below, initiated a number of fields: abelian groups and their characters, L-functions, class number formulae. . . . . The theorem asserts that an arithmetic progression c, c + q, c + 2q, c + 3q, . . . contains infinitely primes if and only if there is no obvious congruence obstruction. An obvious such obstruction would be say that c and q are both even or more generally that the greatest common divisor (c, q) of c and q is bigger than 1. Stated somewhat differently, let L = 0 be a subgroup of Z, so L = qZ for some q ≥ 1, and let O = c+L be the corresponding orbit of c under L, then O contains infinitely many primes iff (c, q) = 1 (strictly speaking this statement is slightly weaker since Dirichlet considers one-sided progressions and here and elsewhere we allow negative numbers and call −p a prime if p is a positive prime). (3) Initial Generalizations: There are at least two well known generalizations of Dirichlet’s Theorem that have been investigated. The first is the generalization of his L-functions to ones associated with general automorphic forms on linear groups. This topic is one of the central themes of modern number theory but other than pointing out that these are used indirectly in proving some of the results mentioned below, I will not discuss them in this lecture. The second generalization is to consider other polynomials besides linear ones. Let f ∈ Z[x] be a polynomial with integer coefficients and let O = c + L as above. Does f(O) contain infinitely many primes? For example if O = Z; is f(x) = x + 1 a prime number for infinitely many x (a question going back at least to Euler). Is f(x) = x(x+2) a product of two primes infinitely often? (this is a reformulation of the twin prime question). Neither of these questions have been answered and the answer to both is surely, yes. We will mention later what progress has been made towards them. In his interesting and provocative article “Logical Dreams” [Sh], Shelah puts forth the dream, that this question of Euler “cannot be decided”. This is rather far fetched but for the more general questions about primes and saturation on very sparse orbits associated with tori that are discussed below, such a possibility should be taken seriously. We turn first in the next paragraph to several variables, that being the setting in which some problems of this type have been resolved. (4) Two Variables: Let O = Z and let f be a nonconstant polynomial in Z[x1, x2]. If f is irreducible in Q[x1, x2] and the greatest common divisor of the numbers f(x) with x ∈ O is 1, then it is conjectured that f takes on infinitely many prime values. In this higher dimensional setting we have found it more intrinsic and natural from many points of view to ask for more. That is the set of x ∈ O at which f(x) is prime should not only produce an infinite set of primes for the values f(x) but these (infinitely) many points should not satisfy any nontrivial Equidistribution and Primes 3 algebraic relation. In the language of algebraic geometry, these points should be Zariski dense in the affine plane A. The Zariski topology on affine n-space A is gotten by declaring the closed sets to be the zero sets (over C) of a system of polynomial equations. Thus a subset S of A is Zariski dense in A iff S is not contained in the zero set of a nontrivial polynomial g(x1, . . . , xn). In A 1 a set is the zero set of a nontrivial polynomial iff the set is finite. So the Zariski dense subsets of A are simply the infinite sets. We denote the operation of taking the Zariski closure of a set in A by Zcl. All the approaches to the conjecture that we are discussing involve giving lower bounds for the number of points in finite subsets of O at which f(x) is prime. Usually one defines these sets by ordering by size of the numbers (so a big box in A) but in some variations of these problems that I discuss later quite different orderings are employed. A measure of the quality of the process is whether in the end the lower bound is strong enough to ensure the Zariski density of the points produced. As far as the conjecture that under the assumptions on f at the beginning of (4), the set of x ∈ O at which f(x) is prime, is Zariski dense in A, the following is known: (i) For f linear it follows from Dirichlet’s theorem. (ii) For f of degree two and f non-degenerate (in the sense of not reducing to a polynomial in one variable) it follows from Iwaniec [Iw]. His method uses the combinatorial sieve which we will discuss a bit further on, as well as the Bombieri-A.Vinogradov theorem which is a sharp quantitative version of Dirichlet’s theorem (when counting primes p of size at most x and which are congruent to varying c modulo q, with q as large as x). (iii) A striking breakthrough was made by Friedlander and Iwaniec [F-I]. It follows from their main result that the conjecture is true for f(x1, x2) = x 2 1 + x 4 2. They exploit the structure of this form in that it can be approached by examining primes α = a+ b √−1 in Z[ √−1] with b = z. This was followed by work of Heath-Brown and the results in [H-M] imply that the conjecture is true for any homogeneous binary cubic form. They exploit a similar structure, in that such an f(x1, x2) is of the form N(x1, x2, 0) where N(x1, x2, x3) is the norm form of cubic extension of Q, so that the problem is to produce prime ideals in the latter with one coordinate set to 0. (iv) If f(x1, x2) is reducible then we seek a Zariski dense set of points x ∈ Z at which f(x) has as few as possible prime factors. For polynomials f of the special form f(x) = f1(x)f2(x) · · ·ft(x), with fj(x) = x1 + gj(x2) where gj ∈ Z[x] and gj(0) = 0, it follows from the results in the recent paper of Tao and Ziegler [T-Z] that the set of x ∈ Z at which f(x) is a product of t primes, is Zariski dense in A. Equivalently the set of x at which f1(x), . . . , ft(x) are simultaneously prime, is dense. This impressive result is based on the breakthrough in Green and Tao [G-T1] in particular their transference principle, which is a tool for replacing sets of positive density in the usual setting of Szémeredi type theorems with a set of positive density in the primes. The corresponding positive density theorem is that of Bergelson and Leibman [B-L]. Note that for these fj’s there is no local obstruction to x1 + gj(x2) being simultaneously prime since for a given q ≥ 1 we can choose x1 ≡ 1(q) and x2 ≡ 0(q) (gj(0) = 0). Apparently this is a feature of Equidistribution and Primes 4 these positive density Szemerédi type theorems in that they don’t allow for congruence obstructions.† The above theorem with gj(x2) = (j − 1)x2, j = 1, . . . , t recovers the Green-Tao theorem, that the primes contain arbitrary long arithmetic progressions. From our point of view in paragraph (8) the amusing difference between the “existence of primes in an arithmetic progression” and that of an “arithmetic progression in the primes”, will be minimized as they both fall under the same umbrella. (5) Hardy-Littlewood n-tuple Conjecture: This is concerned with Z and subgroups L of Z acting by translations. If L is such a group denote by r(L) its rank. We assume L = 0 so that 1 ≤ r ≤ n and also that for each j the coordinate function xj restricted to L is not identically zero. For c ∈ Z and O = c + L the conjecture is about finding points in O all of whose coordinates are simultaneously prime. We state it as the following local to global conjecture: HLC: If O = c + L as above then the set of x = (x1, . . . , xn) ∈ O for which the xj ’s are simultaneously prime, is Zariski dense in Zcl(O) iff for each q ≥ 1 there is an x ∈ O such that x1x2 . . . xn ∈ (Z/qZ)∗. Note that the condition on q, which is obviously necessary for the Zariski density, involves only finitely many q (for each given O). Also to be more accurate, the conjecture in [H-L] concerns only the case of r(L) = 1 (which in fact implies the general case). In this case Zcl(O) is a line and the conjecture asserts that there are infinitely many points in x ∈ O for which the n-tuples (x1, x2, . . . xn) are all prime iff there is no local obstruction. We observe that for any r the Zcl(O) is simply a translate of a linear subspace. The main breakthrough on the HLC as stated above is due to I. Vinogradov (1937) in his proof of his celebrated “ternary Goldbach theorem”, that every sufficiently large positive odd number is a sum of 3 positive prime numbers. His approach was based on Hardy and Littlewood’s circle method, a novel sieve and the technique of bilinear estimates, see Vaughan [Va]. It can be used to prove HLC for a non-degenerate L in Z of rank at least 2. Special cases of HLC in higher dimensions are established by Balog in [Ba] and recently Green and Tao [G-T2] made a striking advance. Their result implies HLC for L ≤ Z and r(L) ≥ 2 and L non-degenerate in a suitable sense. Their approach combines Vinogradov’s methods with their transference principle. It makes crucial use of Gowers’ techniques from his proof of Szemerédi’s theorem, and it has close analogies with the ergodic theoretic proofs of Szémeredi’s theorem due to Furstenberg and in particular the work of Host and Kra [H-K]. These ideas have potential to establish HLC for L ≤ Z of rank at least two (and non-degenerate), which would be quite remarkable. (6) Pythagorean Triples: We turn to examples of orbits O in Z of groups acting by matrix multiplication rather than by translations (i.e. addition). By a Pythagorean triple we mean a point x ∈ Z lying on the affine cone C given as {x : F (x) = x1 + x2 − x3 = 0} and †Though the paper “Intersective polynomials and polynomial Szemerédi theorem” by V. Bergelson, A. Leibman and E. Lesigne posted on ArXiv Oct/25/07, begins to address this issue. Equidistribution and Primes 5 for which gcd(x1, x2, x3) = 1. We are allowing xj to be negative though in this example we could stick to all xj > 0, so that such triples correspond exactly to primitive integral right triangles. Let OF denote the orthogonal group of F , that is the set of 3 × 3 matrices g for which F (xg) = F (x) for all x. Let OF (Z) be the group of all such transformations with entries in Z. Some elements of OF (Z) are A1 = ⎡ ⎣ 1 2 2 −2 −1 −2 2 2 3 ⎤ ⎦ , A2 = ⎡ ⎣ 1 2 2 2 1 2 2 2 3 ⎤ ⎦ , A3 = ⎡ ⎣ −1 −2 −2 2 1 2 2 2 3 ⎤ ⎦ . In fact OF (Z) is generated by A1, A2 and A3. It is a big group and one can show that the set of all Pythagorean triples P is the orbit of (3, 4, 5) under OF (Z), i.e. P = (3, 4, 5)OF (Z). Following the lead of Dirichlet, let L be a subgroup of OF (Z) and let O = (3, 4, 5)L be the corresponding orbit of Pythagorean triples. The area A(x) = x1x2/2 of the corresponding triangle is in Q[x1, x2, x3]. We seek triangles in O for which the area has few prime factors. What is the minimal divisibility of the areas of a Zariski dense (in Zcl(O), which for us will be equal to C) set of triples in O? We return to this later on. As a side comment, a similar problem asks which numbers are the square free parts of the areas of Pythagorean triangles in P ? This is the ancient “congruent number problem” about which much has been written especially because of its connection to the question of the ranks of a certain family of elliptic curves. Heegner [Hee] using his precious method for producing rational points on elliptic curves shows that any prime p ≡ 5 or 7 mod 8 is a congruent number. For a given such p the set of triangles realizing p is very sparse but never-the-less is Zariski dense in C. Via the same relation the congruent number problem is connected to automorphic L-functions through the Birch and Swinnerton-Dyer Conjecture (see [Wi]). (7) Integral Apollonian Packings: As a final example before putting forth the general theory we discuss some Diophantine aspects of integral Apollonian packings. Descartes is well known among other things for his describing various geometric facts in terms of his Cartesian coordinates. One such example is the following relation between four mutually tangent circles: