Maynard 2015 small gaps between primes

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proposition_4_2: Maynard's sieve criterion: if the primes have level of distribution theta, then for every admissible k-set at least the ceiling of theta M_k / 2 of the translates n + h_i are prime for infinitely many n, where M_k is a supremum of ratios of integrals of functions on the simplex.

theorem_1_1: Maynard's theorem that for every natural number m the gap p_{n+m} - p_n spanning m consecutive prime gaps is at most a constant times m^3 e^{4m} for infinitely many n, so bounded intervals contain any fixed number of primes infinitely often.

theorem_1_2: Maynard's theorem that for r large in terms of m, among the m-element subsets of any set of r distinct integers, a proportion bounded below in terms of m are sets whose translates are all prime for infinitely many n.

theorem_1_3: Maynard's unconditional theorem that consecutive primes differ by at most 600 infinitely often, proved from the Bombieri-Vinogradov theorem alone.

theorem_1_4: Maynard's conditional theorem that if the primes have level of distribution theta for every theta < 1, then consecutive primes differ by at most 12, and primes two apart by at most 600, infinitely often.

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Maynard, James, Small gaps between primes. Ann. of Math. (2) 181 (2015), no. 1, 383-413, doi:10.4007/annals.2015.181.1.7. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1311.4600), every other right reserved. The copy read for this card is arXiv:1311.4600v3.

Theorem 1.1 proves that liminf_n (p_{n+m} - p_n) << m^3 e^{4m} for every m in N, so bounded gaps hold for every fixed number of primes, not just pairs; the abstract records the explicit consequences liminf(p_{n+1} - p_n) <= 600 unconditionally and, under the Elliott-Halberstam conjecture, liminf(p_{n+1} - p_n) <= 12 and liminf(p_{n+2} - p_n) <= 600 (Theorems 1.3 and 1.4). Theorem 1.2 shows that, for r large in terms of m, among the m-element subsets of any set of r distinct integers a proportion >>m 1 consists of sets {h_1, ..., h_m} with n + h_1, ..., n + h_m all prime for infinitely many n, which the paper reads as a positive proportion of admissible m-tuples satisfying the prime m-tuples conjecture "in an appropriate sense" (p. 2). The method is a refinement of the Goldston-Pintz-Yildirim sieve using a multidimensional class of weights, which removes the theta = 1/2 barrier in the level of distribution and so needs only Bombieri-Vinogradov rather than Zhang's stronger equidistribution result. The paper notes that Tao independently obtained Theorem 1.1 with a slightly weaker bound by a similar method, and that the results extend to primes in short intervals, in arithmetic progressions, and to simultaneous prime values of linear forms. This bears on problem 6 (infinitely many n with d_n < d{n+1} < d_{n+2}) only as input: Banks, Freiberg and Turnage-Butterbaugh apply the Maynard-Tao theorem for admissible tuples of linear forms, which this paper states as a remark in its introduction (p. 2); for shifts n + h_i it is Proposition 4.2 with the large-k step (4.5) of Section 4 (every admissible k-set with k >= C m^2 e^{4m} has at least m + 1 primes among the n + h_i for infinitely many n). The constant 600 is Theorem 1.3 (k = 105), not Theorem 1.1 with m = 1, which gives only liminf(p_{n+1} - p_n) << e^4 with an unspecified constant.

Source: https://arxiv.org/abs/1311.4600.

Bears on. #6: the paper does not decide the question, which asks for infinitely many n with d_n < d_{n+1} < d_{n+2}; its results give several primes among the translates of an admissible set, not consecutive primes with ordered gaps. Banks, Freiberg and Turnage-Butterbaugh answer it yes using the Maynard-Tao theorem for admissible tuples of linear forms, in Granville's formulation, as input. Its shift case is Proposition 4.2 with the large-k step (4.5). For linear forms the paper has only the unnumbered remark on p. 2, stated for positive integer coefficients and without a separate proof.

Results. Pages are the printed pages of arXiv:1311.4600v3.

• Theorem 1.1 (p. 2): for every m in N, liminf_n (p_{n+m} - p_n) << m^3 e^{4m}.
• Theorem 1.2 (p. 2): for r sufficiently large depending on m and any set A of r distinct integers, the m-subsets {h_1, ..., h_m} of A with n + h_1, ..., n + h_m all prime for infinitely many n make up a proportion >>_m 1 of all m-subsets of A.
• Theorem 1.3 (p. 3): liminf (p_{n+1} - p_n) <= 600.
• Theorem 1.4 (p. 3): if the primes have level of distribution theta for every theta < 1, then liminf (p_{n+1} - p_n) <= 12 and liminf (p_{n+2} - p_n) <= 600.
• Proposition 4.2 (p. 5): if the primes have level of distribution theta > 0, then for every admissible {h_1, ..., h_k} at least ceil(theta M_k / 2) of the n + h_i are prime for infinitely many n; Proposition 4.3 (p. 6) gives M_5 > 2, M_105 > 4 and M_k > log k - 2 log log k - 2 for large k.

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