Goldston 2009 primes tuples i

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theorem_1: If the primes have level of distribution theta > 1/2, then every admissible k-tuple with k >= C(theta) contains at least two primes infinitely often, with k >= 6 sufficing when theta >= 0.971; under Elliott-Halberstam this gives p_{n+1} - p_n <= 16 infinitely often.

theorem_2: Unconditionally, the lower limit of (p_{n+1} - p_n)/log p_n as n tends to infinity is 0, where p_n is the nth prime.

theorem_3: If the primes have level of distribution theta, then for r >= 2 the lower limit E_r of (p_{n+r} - p_n)/log p_n is at most (sqrt r - sqrt(2 theta))^2, and unconditionally E_r <= (sqrt r - 1)^2 for r >= 1.

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Goldston, Daniel A. and Pintz, János and Yıldırım, Cem Y., Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819-862. https://doi.org/10.4007/annals.2009.170.819

The paper introduces the GPY method for showing that primes come close together, driven by the level of distribution theta of primes in arithmetic progressions: the Bombieri-Vinogradov bound (1.3) holding with Q = N^{theta-eps} for every A > 0 and eps > 0 (p. 2), so that theta = 1/2 is known and the Elliott-Halberstam conjecture is theta = 1. Theorem 1 (p. 2) shows that if theta > 1/2 there is an explicitly calculable C(theta) such that every admissible k-tuple with k >= C(theta) contains at least two primes infinitely often, and that k >= 6 suffices when theta >= 0.971; since (n, n+4, n+6, n+10, n+12, n+16) is admissible, the Elliott-Halberstam conjecture implies liminf (p_{n+1} - p_n) <= 16, display (1.7). Theorem 2 (p. 2) is unconditional: E_1 = liminf (p_{n+1} - p_n)/log p_n = 0, so consecutive primes are infinitely often closer than any fixed positive multiple of the average spacing. Its proof averages the tuple-detecting weight, a truncated divisor sum of (n+h_1)...(n+h_k), over all k-tuples of shifts in an interval, which also gives E_r <= max(r - 2 theta, 0), display (1.11). Theorem 3 (p. 4) sharpens this to E_r <= (sqrt r - sqrt(2 theta))^2 for r >= 2, and unconditionally E_r <= (sqrt r - 1)^2 for r >= 1, where E_r is the lower limit of (p_{n+r} - p_n)/log p_n. The abstract says the last unconditional result will be considerably improved in a later paper.

Source: https://arxiv.org/abs/math/0508185. The copy read for this card is the arXiv preprint (v1, 10 August 2005). The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0508185), every other right reserved.

Results. Labels and pages are those of the arXiv preprint named above.

• Theorem 1 (p. 2), with display (1.7): under level of distribution theta > 1/2, every admissible k-tuple with k >= C(theta) contains two primes infinitely often, k >= 6 sufficing for theta >= 0.971; the Elliott-Halberstam conjecture implies p_{n+1} - p_n <= 16 infinitely often.
• Theorem 2 (p. 2): unconditionally, liminf (p_{n+1} - p_n)/log p_n = 0.
• Theorem 3 (p. 4): under level of distribution theta, E_r <= (sqrt r - sqrt(2 theta))^2 for r >= 2; unconditionally E_r <= (sqrt r - 1)^2 for r >= 1.

Read status. Claims checked for the three results above, read clause by clause on the print; their proofs, in Sections 3 and 10, were read for their structure only, and Propositions 1 and 2 (pp. 7-8), on which all three rest, were not checked.

Bears on.

• Problem 5: asks whether every C >= 0 is the limit of (p_{n_i+1} - p_{n_i})/log n_i along some sequence n_i. Theorem 2, with log p_n ~ log n, answers yes for C = 0; the paper says nothing about any C > 0.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.