Bruedern 2017 local oscillations moderately dense sequences primes

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theorem_1: Brüdern and Elsholtz's theorem that if a set of primes P has #{p in P : p <= x} (log x)^{4/3}/x tending to infinity, and p_n enumerates P increasingly, then p_{n+1}^2 - p_n p_{n+2} changes sign infinitely often.

theorem_2: Brüdern and Elsholtz's two-sided bound for the curvature K_N(P) of a delta-dense set of primes in a progression: at most 500 delta_N^{-1} log N for N >= N_0(q), and at least 10^{-8} delta_N^3 log N when delta(x)^2 log x tends to infinity; for a whole progression both bounds are of order log N.

theorem_3: Brüdern and Elsholtz's bounds for the sum over N < n <= 2N of |p_{n+2} - 2p_{n+1} + p_n|/p_n for a delta-dense set of primes: at most 11/delta_{2N+2} for N >= N_0(q), and at least 10^{-7} delta_{2N}^3 when delta(x)^2 log x tends to infinity; a scattered example with delta(x) = 1/log x has a single term of the order of the upper bound.

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Joerg Bruedern, Christian Elsholtz, Local oscillations in moderately dense sequences of primes. arXiv preprint (2017). arXiv:1702.00289.

Theorem 1 proves that if a set P of primes satisfies #{p in P : p <= x} (log x)^(4/3) / x tending to infinity, and p_n enumerates P, then p_{n+1}^2 - p_n p_{n+2} changes sign infinitely often, extending the Erdos-Turan result from the full sequence of primes. The main object is Renyi's curvature K_N(P), the total turning of the polygonal line through the points z_n = n + i log p_n; unboundedness of the curvature forces the sign changes. Theorem 2 gives two-sided curvature estimates for delta-dense subsets of an arithmetic progression P_{q,a}, where delta is decreasing with delta(x) >= 1/log x: for N >= N_0(q), K_N(P) <= 500 delta_N^{-1} log N always, and K_N(P) >= 10^{-8} delta_N^3 log N when delta(x)^2 log x tends to infinity; with delta = 1 this contains the Erdos-Renyi order of magnitude log N << K_N << log N for the full sequence of primes. The curvature estimates use only lower bounds for the counting function of P, where Erdos and Renyi used the prime number theorem. Theorem 3 bounds the sums of |p_{n+2} - 2p_{n+1} + p_n|/p_n over N < n <= 2N above and below; the proof of Theorem 2 uses its upper-bound argument and Lemma 4, a more explicit form of its lower bound. Read status: claims checked for Theorems 1 to 3 and the Corollary, read clause by clause on the print; the proofs were read for structure only.

Source: https://arxiv.org/abs/1702.00289. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1702.00289), every other right reserved.

Bears on. #455: no result. The problem's condition of non-decreasing gaps says that the second differences p_{n+2} - 2p_{n+1} + p_n are never negative, the quantity Theorem 3 bounds; but Richter's bound liminf q_n/n^2 > 0 for such a sequence q_n, recorded on the problem page, gives it O(sqrt(x)) terms up to x, far below the density that every theorem here assumes, so none of them applies to it. The paper does not cite Richter.

Results.

• Theorem 1 (p. 1): if #{p in P : p <= x}(log x)^(4/3)/x tends to infinity and p_n enumerates P in increasing order, then p_{n+1}^2 - p_n p_{n+2} changes sign infinitely often.
• Theorem 2 (p. 2) and its Corollary (p. 3): for x_0 >= 3, delta decreasing with delta(x) >= 1/log x, and N >= N_0(q), every delta-dense set P of primes in some P_{q,a} has K_N(P) <= 500 delta_N^{-1} log N, and K_N(P) >= 10^{-8} delta_N^3 log N when delta(x)^2 log x tends to infinity; for P = P_{q,a}, 10^{-8} log N <= K_N <= 500 log N.
• Theorem 3 (p. 3): under the same hypotheses the sum of |Delta_n|/p_n over N < n <= 2N is at most 11/delta_{2N+2}, and at least 10^{-7} delta_{2N}^3 when delta(x)^2 log x tends to infinity; the scattered set of Section 5 (pp. 12--13), delta-dense with delta(x) = 1/log x, has for suitable large N a single term of that sum exceeding (1/3) log N, which the paper calls of the order of delta_{2N+2}^{-1}.

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