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Bounded gaps between primes. (English) Zbl 1290.11128

In this historic paper it is proved for the first time that there is an (even) integer \(h\) which occurs infinitely often as the difference between consecutive primes. Indeed it is shown that there is such an \(h\) below \(7\times 10^7\). It is conjectured of course that \(h=2\) is admissible, but the above statement is our closest approximation yet to the twin prime conjecture.
The proof builds on the work of D. A. Goldston et al. [Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096)] who showed that \[ \liminf_{n\rightarrow\infty}\frac{p_{n+1}-p_n}{\log p_n}=0. \] The GPY method compares the sums \[ \sum_{x<n\leq 2x}w(n)^2 \] and \[ \sum_{x<n\leq 2x}\left(\sum_{i=1}^k\Lambda(n+h_i)\right)w(n)^2 \] where \(h_1,\ldots,h_k\) is a fixed “admissible” set, and \[ w(n)=\sum_{d\mid P(n),\,d<D}\lambda_d \] with \(P(n)=\prod_{_i=1}^k(n+h_i)\). To calculate the second sum one requires the primes to have level of distribution at least \(D^2\), so that the sieve coefficients \(\lambda_d\) must be supported essentially on \([1,x^{1/4-\varepsilon}]\) for the Bombieri–Vinogradov Theorem to apply. Goldston, Pintz and Yıldırım showed that if one could extend the Bombieri-Vinogradov Theorem to give a level of distribution \(x^{\theta}\) for the primes up to \(x\), with some constant \(\theta>\tfrac12\), then one could prove the bounded gaps result.
Zhang’s argument uses two key ideas. Firstly, he is able to establish a suitable extension of the Bombieri–Vinogradov Theorem, in which the moduli are restricted to appropriately smooth integers. Secondly, he observes that if one insists that the sieve coefficients \(\lambda_d\) are restricted to suitably smooth integers the loss in the GPY method is surprisingly small. In fact this idea already occurs explicitly in work of Y. Motohashi and J. Pintz [Bull. Lond. Math. Soc. 40, No. 2, 298–310 (2008; Zbl 1278.11090)], but the author seems not to have been aware of this.
The author’s extension of the Bombieri–Vinogradov Theorem is essentially as follows. Let \(P\) be the product of the primes \(p<x^{\varpi}\), where \[ \varpi=\frac{1}{1168}. \] (Any sufficiently small value will do.) For \(i\leq k\) let \[ \mathcal{C}_i(d)=\{c\in\mathbb{N}:\, c\leq d,\, (c,d)=1,\, d\mid P(c-h_i)\}. \] Then for \(i\leq k\) we have \[ \sum_{d<x^{1/2+2\varpi},\;d\mid P} \sum_{c\in\mathcal{C}_i(d)}\left|\psi(x;d,c)-\frac{x}{\varphi(d)}\right|\ll_A x(\log x)^{-A} \] for any fixed \(A>0\). The reader should note that the sum over \(c\) will include the residue classes of \(c_i-c_j\) whenever these are coprime to \(d\). These classes are independent of \(d\). However when \(d\) is composite there may also be residue classes which depend on \(d\). The necessity to treat such classes was a difficulty in earlier attacks on the bounded gaps problem.
In order to prove the extension of the Bombieri–Vinogradov Theorem the paper uses the reviewer’s generalised Vaughan identity, producing various types of convolutions of arithmetic functions. There are Type I sums, which are easily handled. There are Type II sums involving \(a*b\) with \(a\) supported on \((x^{3/8+8\varpi},2x^{1/2})\). These are handled in two distinct ways, where the support is in \((x^{3/8+8\varpi},x^{1/2-4\varpi}]\) and \((x^{1/2-4\varpi},2x^{1/2})\) respectively. (The paper calls these “Type I” and “Type II”, somewhat confusingly.) Finally there are sums involving \(a*c_1*c_2*c_3\), where \(c_1,c_2,c_3\) are the characteristic functions of intervals \((N_i,2N_i]\) such that \(N_3\leq N_2\leq N_1\) and \(N_2N_3\geq x^{5/8-8\varpi}\). These are dubbed “Type III” sums.
The estimations of the “Type I” and “Type II” sums (in the author’s parlance) are motivated by the works of É. Fouvry and H. Iwaniec [Acta Arith. 42, 197–218 (1983; Zbl 0517.10045)] and E. Bombieri et al. [Acta Math. 156, 203–251 (1986; Zbl 0588.10042)], concerning extensions of the range in the Bombieri–Vinogradov Theorem. However the only input on Kloosterman sums that is required is the classical Weil bound. The treatment of the “Type III” sums is based on the work of J. B. Friedlander and H. Iwaniec [Ann. Math. (2) 121, 319–350 (1985; Zbl 0572.10029)] on \(d_3(n)\) in arithmetic progressions. The argument given by Friedlander and Iwaniec is insufficient on its own, but an extra saving is obtained using the fact that the moduli are smooth. The exponential sum estimate of Birch and Bombieri, given in the appendix to the paper by Friedlander and Iwaniec, is crucial.
The techniques in this paper have been thoroughly investigated and extended by the polymath8 project [W. Castryck et al., Algebra Number Theory 8, No. 9, 2067–2199 (2014; Zbl 1307.11097)]. However a slightly different approach due to J. Maynard [Ann. Math. (2) 181, No. 1, 383–413 (2015; Zbl 1306.11073)] (and also Tao, unpublished) avoids extensions of the Bombieri–Vinogradov Theorem, and indeed proves not only that \[ \liminf_{n\rightarrow\infty} p_{n+1}-p_n\leq 600, \] but also that \[ \liminf_{n\rightarrow\infty} p_{n+h}-p_n<\infty \] for any fixed \(m\in\mathbb{N}\). In spite of these developments the present paper remains an outstanding landmark in the history of prime number theory. Moreover the author’s novel extension of the Bombieri–Vinogradov Theorem has potential applications beyond the bounded prime gaps question.

MSC:

11N05 Distribution of primes
11N13 Primes in congruence classes
11N35 Sieves
11N36 Applications of sieve methods
11L07 Estimates on exponential sums
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