Pintz 2016 polignac numbers conjectures erdos gaps primes

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main_theorem: Pintz's Main Theorem: for k >= 3.5 x 10^6, every admissible k-tuple in [0, eps log N] has, for at least c_2(k) S(H) N/log^k N integers n in [N,2N), two consecutive primes among the n + h_i and every prime factor of every n + h_i greater than n^{c_1(k)}; it is the input to Theorems 1 to 6.

theorem_1: Pintz's theorem that there is an explicitly calculable constant c such that, for N > N_0, at least cN Polignac numbers lie below N, an even number 2k being a Polignac number when p_{n+1} - p_n = 2k for infinitely many n.

theorem_2: Pintz's theorem that there is an ineffective constant C' such that every interval [M, M + C'] contains at least one Polignac number, an even number occurring as a gap between consecutive primes infinitely often.

theorem_3: Pintz's theorem that there is an ineffective constant c > 0 with [0,c] contained in the set J of limit points of (p_{n+1} - p_n)/log n, a weaker form of Erdős's conjecture that J is all of [0, infinity].

theorem_4: Pintz's theorem that for every slowly oscillating f with f(n) <= log n and f(n) tending to infinity there is an ineffective c_f > 0 such that [0,c_f] is contained in the set of limit points of (p_{n+1} - p_n)/f(n).

theorem_5: Pintz's theorem that the ratio of consecutive prime gaps d_{n+1}/d_n satisfies liminf (d_{n+1}/d_n) log n < infinity and limsup (d_{n+1}/d_n)/log n > 0, a strong form of Erdős's conjecture that the liminf of the ratio is 0 and its limsup is infinity.

theorem_6: Pintz's theorem that some d <= 7 x 10^7 admits arbitrarily long arithmetic progressions of primes p for each of which p + d is the prime following p, combining Zhang's method with the Green-Tao theorem.

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Pintz, János, Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture. From arithmetic to zeta-functions (2016), 367-384. doi:10.1007/978-3-319-28203-9_22. The copy read for this card is the arXiv version (arXiv:1305.6289v1), whose record names arXiv's non-exclusive distribution license, every other right reserved.

Building on Zhang's bounded gap theorem and the Goldston-Pintz-Yildirim sieve, Pintz derives a series of unconditional results on prime gaps from one strengthened tuple theorem, his Main Theorem (p. 6): for k >= 3.5 x 10^6, the paper's Conjecture DHL*(k,2) holds, so every admissible k-tuple contained in [0, eps log N], eps sufficiently small, has, for at least c_2(k) S(H) N/log^k N integers n in [N,2N), two consecutive primes among the n + h_i and every n + h_i free of prime factors up to n^{c_1(k)}. Theorem 1 (p. 3) shows the strong Polignac numbers (even 2k with p_{n+1} - p_n = 2k infinitely often) have positive lower asymptotic density, and Theorem 2 (p. 3) shows every interval [M, M+C'] contains a Polignac number for an ineffective constant C'. Theorem 3 (p. 4) gives what the paper calls a weaker form of Erdős's 1955 conjecture that the set J of limit points of d_n/log n is all of [0, infinity]: there is an ineffective c > 0 with [0,c] contained in J. Theorem 4 (p. 4) extends this to every slowly oscillating f with f(n) <= log n and f(n) -> infinity, giving an ineffective c_f > 0 with [0,c_f] inside the set of limit points of d_n/f(n). The paper also recalls earlier small-gap results: Goldston, Pintz and Yildirim's liminf d_n/((log n)^{1/2}(log log n)^2) < infinity (Theorem D, p. 3) and the author's improvement of the exponent to 3/7, liminf d_n/((log n)^{3/7}(log log n)^{4/7}) < infinity (Theorem E, p. 3), cited from his Turán Memorial paper and not proved here. Theorem 5 (p. 4) proves Erdős's conjectures liminf d_{n+1}/d_n = 0 and limsup d_{n+1}/d_n = infinity in a stronger form (liminf (d_{n+1}/d_n) log n < infinity and limsup (d_{n+1}/d_n)/log n > 0), and Theorem 6 (p. 5) gives a d <= 7 x 10^7 with arbitrarily long arithmetic progressions of primes p whose next prime is p + d. The proofs of the Main Theorem and of Theorems 1 and 6 lean on earlier works and are described rather than written out; Section 8 (p. 12) sketches how all the results can be made effective.

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv:1305.6289v1; labels and pages are that version's. No proof is checked step by step.

Source: https://arxiv.org/abs/1305.6289.

Bears on.

• #5: Theorem 3 answers the problem's question yes for every C in [0,c], where c > 0 is ineffective and not determined by the paper: each such C is the limit of (p_{n_i+1} - p_{n_i})/log n_i along some sequence n_i. It says nothing about C > c. Theorem 4 contains Theorem 3 as the case f(n) = log n.

Results.

• Main Theorem (p. 6): Conjecture DHL*(k,2) holds for k >= 3.5 x 10^6.
• Theorem 1 (p. 3): There is an explicitly calculable constant c such that for N > N_0 at least cN Polignac numbers lie below N; Polignac numbers have positive lower asymptotic density.
• Theorem 2 (p. 3): There is an ineffective constant C' such that every interval [M, M+C'] contains at least one Polignac number.
• Theorem 3 (p. 4): There is an ineffective c > 0 such that [0,c] is contained in the set J of limit points of d_n/log n.
• Theorem 4 (p. 4): For every slowly oscillating f with f(n) <= log n and f(n) -> infinity there is an ineffective c_f > 0 with [0,c_f] contained in the set of limit points of d_n/f(n).
• Theorem 5 (p. 4): liminf (d_{n+1}/d_n) log n < infinity and limsup (d_{n+1}/d_n)/log n > 0.
• Theorem 6 (p. 5): There is a d <= 7 x 10^7 with arbitrarily long arithmetic progressions of primes p for each of which p + d is the next prime.

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