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New Equidistribution Estimates of Zhang Type NEW EQUIDISTRIBUTION ESTIMATES OF ZHANG TYPE D.H.J. POLYMATH Abstract. We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese 1 7 Remainder Theorem conditions, obtaining an exponent of distribution 2 ` 300 . Contents 1. Introduction 1 2. Preliminaries 8 3. Applying the Heath-Brown identity 14 4. One-dimensional exponential sums 23 5. Type I and Type II estimates 38 6. Trace functions and multidimensional exponential sum estimates 59 7. The Type III estimate 82 8. An improved Type I estimate 94 References 105 1. Introduction In May 2013, Y. Zhang [52] proved the existence of infinitely many pairs of primes with bounded gaps. In particular, he showed that there exists at least one h ¥ 2 such that the set tp prime | p ` h is primeu is infinite. (In fact, he showed this for some even h between 2 and 7 ˆ 107, although the precise value of h could not be extracted from his method.) Zhang's work started from the method of Goldston, Pintz and Yıldırım [23], who had earlier proved the bounded gap property, conditionally on distribution estimates arXiv:1402.0811v3 [math.NT] 3 Sep 2014 concerning primes in arithmetic progressions to large moduli, i.e., beyond the reach of the Bombieri{Vinogradov theorem. Based on work of Fouvry and Iwaniec [11, 12, 13, 14] and Bombieri, Friedlander and Iwaniec [3, 4, 5], distribution estimates going beyond the Bombieri{Vinogradov range for arithmetic functions such as the von Mangoldt function were already known. However, they involved restrictions concerning the residue classes which were incompatible with the method of Goldston, Pintz and Yıldırım. Zhang's resolution of this difficulty proceeded in two stages. First, he isolated a weaker distribution estimate that sufficed to obtain the bounded gap property (still 2010 Mathematics Subject Classification. 11P32. 1 2 D.H.J. POLYMATH involving the crucial feature of going beyond the range accessible to the Bombieri{ Vinogradov technique), where (roughly speaking) only smooth1 moduli were involved, and the residue classes had to obey strong multiplicative constraints (the possibility of such a weakening had been already noticed by Motohashi and Pintz [43]). Secondly, and more significantly, Zhang then proved such a distribution estimate. This revolutionary achievement led to a flurry of activity. In particular, the Poly- math8 project was initiated by T. Tao with the goal first of understanding, and then of improving and streamlining, where possible, the argument of Zhang. This was highly successful, and through the efforts of a number of people, reached a state in October 2013, when the first version of this paper [46] established the bounded gap property in the form lim infppn`1 ´ pnq ¤ 4680; where pn denotes the n-th prime number. However, at that time, J. Maynard [38] obtained another conceptual breakthrough, by showing how a modification of the structure and of the main term analysis of the method of Goldston, Pintz and Yıldırım was able to establish not only the bounded gap property using only the Bombieri-Vinogradov theorem (in fact the bound lim infppn`1 ´ pnq ¤ 600 obtained was significantly better than the one obtained by Polymath8), but also the bounds lim infppn`k ´ pnq ă `8 for any fixed k ¥ 1 (in a quantitative way), something which was out of reach of the earlier methods, even for k “ 2. (Similar results were obtained independently in unpublished work of T. Tao.) Because of this development, a part of the Polymath8 paper became essentially obsolete. Nevertheless, the distribution estimate for primes in arithmetic progressions are not superceded by the new method, and it has considerable interest for analytic number theory. Indeed, it is the best known result concerning primes in arithmetic progressions to large moduli without fixing the residue class. (When the class is fixed, the best results remain those of Bombieri, Friedlander and Iwaniec, improving on those of Fouvry and Iwaniec [13, 3].) The results here are also needed to obtain the best known bounds on lim infppn`k ´ pnq for large values of k; see [47]. The present version of the work of Polymath8 therefore contains only the state- ment and proof of these estimates. We note however that some of the earlier version is incorporated in our subsequent paper [47], which builds on Maynard's method to further improve many bounds concerning gaps between primes, both conditional and unconditional. Furthermore, the original version of this paper, and the history of its elaboration, remain available online [46]. Our main theorem is: Theorem 1.1. Let θ “ 1{2 ` 7{300. Let " ¡ 0 and A ¥ 1 be fixed real numbers. For all primes p, let ap be a fixed invertible residue class modulo p, and for q ¥ 1 squarefree, denote by aq the unique invertible residue class modulo q such that aq ” ap modulo all primes p dividing q. 1 I.e, friable. EQUIDISTRIBUTION OF PRIMES IN ARITHMETIC PROGRESSIONS 3 There exists δ ¡ 0, depending only on ", such that for x ¥ 1, we have x x px; q; a q ´ ! ; q 'pqq plog xqA q¤xθ´" ˇ ˇ q xδ-smooth,¸ squarefreeˇ ˇ ˇ ˇ where the implied constant depends only on A, " and δ, and in particular is independent of the residue classes papq. In this statement, we have, as usual, defined px; q; aq “ Λpnq; n¤x n“¸a pqq where Λ is the von Mangoldt function. In [52], Zhang establishes a weaker form of Theorem 1.1, with θ “ 1{2`1{584, and with the aq required to be roots of a polynomial P of the form P pnq :“ 1¤j¤k;j‰ipn ` hj ´ hiq for a fixed admissible tuple ph1; : : : ; hkq and i “ 1; : : : ; k. In fact, we will prove± a number of variants of this bound. These involve either weaker restrictions on the moduli (\dense-divisibility", instead of smoothness, which may be useful in some applications), or smaller values of θ ¡ 1{2, but with significantly simpler proofs. In particular, although the full strength of Theorem 1.1 depends in crucial ways on applications of Deligne's deepest form of the Riemann Hypothesis over finite fields, we show that, for a smaller value of θ ¡ 1{2, it is possible to obtain the same estimate by means of Weil's theory of exponential sums in one variable over finite fields. The outline of this paper is as follows: in the next section, we briefly outline the strategy, starting from the work of Bombieri, Fouvry, Friedlander, and Iwaniec (in chronological order, [12, 13, 21, 3, 4, 5, 14]), and explaining Zhang's innovations. These involve different types of estimates of bilinear or trilinear nature, which we present in turn. All involve estimates for exponential sums over finite fields. We therefore survey the relevant theory, separating that part depending only on one-variable character sums of Weil type (Section 4), and the much deeper one which depends on Deligne's form of the Riemann Hypothesis (Section 6). In both cases, we present the formalism in sometimes greater generality than strictly needed, as these results are of independent interest and might be useful for other applications. 1.1. Overview of proof. We begin with a brief and informal overview of the methods used in this paper. Important work of Fouvry and Iwaniec [12, 13] and of Bombieri, Friedlander and Iwaniec [3, 4, 5] had succeeded, in some cases, in establishing distribution results similar to Theorem 1.1, in fact with θ as large as 1{2 ` 1{14, but with the restriction that the residue classes ap are obtained by reduction modulo p of a fixed integer a ¥ 1. Following the techniques of Bombieri, Fouvry, Friedlander and Iwaniec, Zhang used the Heath-Brown identity [26] to reduce the proof of (his version of) Theorem 1.1 to the verification of three families of estimates, which he called \Type I", \Type II", and \Type III". These estimates were then reduced to exponential sum estimates, using techniques such as Linnik's dispersion method, completion of sums, and Weyl differenc- ing. Ultimately, the exponential sum estimates were established by applications of the Riemann Hypothesis over finite fields, in analogy with all previous works of this type. The final part of Zhang's argument is closely related to the study of the distribution of 4 D.H.J. POLYMATH the ternary divisor function in arithmetic progressions by Friedlander and Iwaniec [21], and indeed the final exponential sum estimate that Zhang uses already appears in their work (this estimate was proved by Birch and Bombieri in the Appendix to [21]). An im- portant point is that, by using techniques that are closer to some older work of Fouvry and Iwaniec [12], Zhang avoids using the spectral theory of automorphic forms, which is a key ingredient in [13] and [3], and one of the sources of the limitation to a fixed residue in these works. Our proof of Theorem 1.1 follows the same general strategy as Zhang's, with improve- ments and refinements. First, we apply the Heath-Brown identity [26] in Section 3, with little change com- pared with Zhang's argument, reducing to the \bilinear" (Types I/II) and \trilinear" (Type III) estimates. For the Type I and Type II estimates, we follow the arguments of Zhang to reduce to the task of bounding incomplete exponential sums similar to c n¯ ` c n ` l e 1 2 ; q Năn¤2N ¸ ´ ¯ (where epzq “ e2iπz andx ¯ denotes the inverse of x modulo q) for various parameters N; c1; c2; l; q. We obtain significant improvements of Zhang's numerology at this stage, by exploiting the smooth (or at least densely divisible) nature of q, using the q-van der Corput A-process of Heath-Brown [25] and Graham-Ringrose [24], combined with the Riemann Hypothesis for curves over finite fields.
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