NOTES ON ZHANG S PRIME GAPS PAPER

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1 NOTES ON ZHANG S PRIME GAPS PAPER TERENCE TAO. Zhang s results For any natural number H, let P (H) denote the assertion that there are infinitely many pairs of distinct primes p, q with p q H; thus for instance P (2) is the twin prime conjecture. In [5], Zhang established for the first time a result of the form P (H) for some finite H: Theorem. [5] P (7 0 7 ) is true. This result is deduced from the following result. Call an admissible set to be a finite set of integers H which avoids at least one residue class modulo p for each prime p. For any natural number, let Q( ) denote the assertion that for any admissible set H of integers of cardinality, there are infinitely many translates n + H of H that contain at least two primes. Note that if H is an admissible set of cardinality, then Q( ) implies P (diam(h)). Theorem is then derived from Theorem 2 below, together with a construction of an admissible set of cardinality and diameter at most : Theorem 2. [5, Theorem ] Q( ) is true. One can improve Theorem by constructing narrower admissible sets H of the specified cardinality ; in particular one can show P (57, 554, 086) in this fashion [2], by selecting a set H of the form H := {±, ±p m,..., ±p m+k0/2 } with := and m := 36, 76, with p n denoting the n th prime; this construction first appears in [4]. Theorem 2 is in turn primarily deduced from a deep improvement of the Bombieri- Vinogradov inequality, which we now pause to state. For any ϖ > 0, let R(ϖ) denote the assertion that the estimate (θ; d, c) x log A x d<d 2 ;d P c h 0+C(d) for all admissible tuples H, all h 0 H, all fixed A > 0 and all sufficiently large x, D := x /4+ϖ D := x ϖ P := p<d p, 99 Mathematics Subject Classification.???

2 2 TERENCE TAO and for each positive integer d, C(d) is the set of residue classes c mod d coprime to d such that h H (c + h) = 0 mod d. Also, θ : N R is the function with θ(n) := log n when n is prime, and θ(n) = 0 otherwise, and (θ; d, c) := θ(n) θ(n) φ(n) x n 2x:n=c mod d x n 2x;(n,d)= is the error term in the prime number theorem in arithmetic progressions for c mod d. Here the implied constant can depend on H, A but is independent of x. Theorem 3. [5, Theorem 2] R( 68 ) is true. Any result of the form R(ϖ) for some ϖ > 0 implies a result of the form Q( ) for some <. We review this argument (a form of which already appears in [3]) in Section Deducing Theorem 2 from Theorem 3 We now recall how R(ϖ) for some ϖ > 0 can be used to establish Q( ) for some <. Let ϖ be given, and let and l 0 be large integer parameters to be chosen later. In [5], ϖ = /68, = , and l 0 = 80, but one has the freedom to vary these parameters provided that a certain inequality (4) holds. Let x be a large number, let H be an admissible tuple of cardinality, and let D, D, P be as in the introduction. We introduce the Goldston-Pintz-Yildirim weight function [] λ(n) := ( + l 0 )! d P (n);d D µ(d)(log D d )k0+l0 P is the polynomial P (n) := h H (n + h), and the sums S := λ(n) 2 and If one can show that S 2 := x n 2x x n 2x( h H θ(n + h))λ(n) 2. () S 2 (log 3x)S > 0 for all sufficiently large x, then this implies an infinite number of translates n + H that contain at least two primes, giving Q( ) as required. To establish () we need upper bounds on S and lower bounds on S 2. It turns out that one can establish bounds of the form (2) S + κ ( + 2l 0 )! ( 2l0 l 0 ) Gx(log D) k0+2l0 + o(x log k0+2l0 x) and (3) S 2 k ( ) 0( κ 2 ) 2l0 + 2 Gx(log D) k0+2l0+ + o(x log k0+2l0+ x) ( + 2l 0 + )! l 0 +

3 NOTES ON ZHANG S PRIME GAPS PAPER 3 κ, κ 2 > 0 are two quantities depending on, l 0, ϖ to be defined later, and G is the singular series G := p ( ν p p )( p ) k0 ν p = C(p) is the number of distinct residue classes occupied by H mod p. Except for the error terms κ, κ 2, these bounds are natural from sieve-theoretic considerations, and are unlikely to be improvable without new breakthroughs in sieve theory. It is a standard fact that 0 < G <. Since log x = 4 + log D, we thus obtain () for sufficiently large x as soon as ( ) ( κ 2 ) 2l0 + 2 > ( + 2l 0 + )! l 0 + which simplifies to (4) ( + )( κ 2 ) > ( κ ( + 2l 0 )! ( ) 2l0 l 0 2l 0 + )( + 2l 0 + )( + κ ) If ϖ > 0, this inequality can be satisfied if κ, κ 2 are small enough and, l 0 are large enough. Note that ( + 2l 0 + )( + 2l 0 + ) ( + and so a necessary condition for (4) to be satisfied is that > (( + ) /2 ) 2 ) 2 k /2 0 which places a theoretical limit as to how small a value of one can extract from a given value of ϖ. In particular, with the choice ϖ = /68 from Theorem 3, one cannot hope for a better value of than 34, 640. This is an order of magnitude better than the value = in Theorem 2; this is due to the need to get good bounds on κ, κ 2. There is thus scope to improve a fair bit without hitting the full limits of the Goldston-Pintz-Yildirim method or without improving ϖ by improving the bounds on κ, κ 2. In [5, 4] it is shown that one can take κ = δ ( + δ log( + )) δ := ( + ) k0 ( ) k0 + 2l 0 and δ 2 is any quantity for which one has the upper bound (5) q P ;q<d P := ϱ (q) q δ 2 + o() D p<d p

4 4 TERENCE TAO and ϱ (q) is the multiplicative function on square-free integers with ϱ (p) = ν p for all p; see [5, (4.5)]. Similarly, in [5, 5] it is shown that one can take κ 2 = δ ( + )( + δ2 2 + log( + ( ) )) k0 + 2l 0 + which simplifies to ( + 2l 0 + ) κ 2 = ( + )κ (2l 0 + )(2l 0 + 2). Now we turn to the problem of estimating δ 2. Zhang does this as follows. Firstly, we have ν p for all primes p, so that ϱ (q) k j 0 when q is the product of exactly j primes. Thus one can bound the left-hand side of (5) by k j 0 D p <...<p j<d:p...p j<d p... p j. Note that D = D +, so we can restrict j to j (assuming that is an integer; note that it is equal to 292 in the case ϖ = /68). If we then discard the p... p j < D constraint, we can then bound the above expression by / k j 0 j! ( p )j. D p<d By the prime number theorem we have p = log log D log log D + o() = log( + ) + o() D p<d and so Zhang obtains the value (6) δ 2 := / ( log( + ))j j! for δ 2. This is however a bit wasteful because we can take further advantage of the p... p j < D constraint by the method of Buchstab iteration. For any x, y > 0, we define the quantity (7) Φ(x, y) := k j 0 p... p j y p <...<p j:p...p j<x then we can bound the left-hand side of (5) by Φ(D, D). We observe that (8) Φ(x, y) = when y > x, while in general we have the Buchstab identity (9) Φ(x, y) + p Φ(x p, p) y p<x as can be seen by isolating the smallest prime p in all the terms in (7) with j. (This inequality is very close to being an identity.) We can iterate this identity to obtain the following conclusion:

5 NOTES ON ZHANG S PRIME GAPS PAPER 5 Lemma 4. For any n, we have n Φ(x, y) ( + log( + )) + o() j j= whenever y is large and x y n, with the error o() going to zero as y uniformly in x for fixed n. Proof. Write A n := n j= ( + log( + j )). We prove the bound Φ(x, y) A n + o() by strong induction on n. The case n = follows from (8). Now suppose that n > and that the claim has already been proven for smaller n. Let x y n. Note that x p pj whenever p x j+. We thus have from (9) and the induction hypothesis that n Φ(x, y) + j= x j+ p<x j p (A j + o()); applying the prime number theorem we have p (A j + o()) = A j log( + j ) + o() x j+ p<x j and the claim follows from the telescoping identity n A n = + A j log( + j ). j= From this lemma with n := +, x = D, and y = D we see that we can take δ 2 to be δ 2 = j= ( + log( + j )). This is roughly equal to k / 0 /( )!, which improves over (6) by a factor of about (log( + ))/. References [] D. Goldston, J. Pintz, C. Yildirim, Primes in tuples. I, Ann. of Math. (2) 70 (2009), no. 2, [2] S. Morrison, [3] Y. Motohashi, J. Pintz, A smoothed GPY sieve, Bull. Lond. Math. Soc. 40 (2008), no. 2, [4] I. Richards, D. Hensley, Primes in intervals, Acta Arith. 25 (973/74), [5] Y. Zhang, Bounded gaps between primes, preprint. Department of Mathematics, UCLA, Los Angeles CA address: tao@@math.ucla.edu URL: tao