Lichtman 2022 proof erdos primitive set conjecture

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theorem_1_10: Lichtman's refinement of a 1966 theorem of Erdős, Sárközy and Szemerédi: a set with positive upper log log density contains an infinite L-divisibility chain whose count up to y, divided by log log y, has upper limit at least that density divided by e^gamma.

theorem_1_2: Lichtman's proof of the Erdős primitive set conjecture: for every primitive set A of integers greater than 1, the sum of 1/(a log a) over A is at most the same sum over the primes, 1.6366....

theorem_1_3: Lichtman's theorem that for every primitive set A and every prime p > 2, the sum of 1/(a log a) over the elements of A with least prime factor p is at most 1/(p log p); whether p = 2 has this property is left open.

theorem_1_5: Lichtman's bound toward the 1968 Erdős, Sárközy and Szemerédi conjecture: the limit as x tends to infinity of the supremum of f(A) over primitive sets A contained in [x, infinity) is at most e^gamma pi/4, about 1.399.

theorem_1_8: Lichtman's refinement of the Davenport and Erdős chain theorem: a set of positive integers with positive upper logarithmic density contains an infinite chain in which each term is an L-multiple of the one before.

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Jared Duker Lichtman, A proof of the Erdős primitive set conjecture. arXiv:2202.02384 (2022); published in Forum Math. Pi 11 (2023), e18. The copy read for this card is arXiv v4 (25 December 2024), whose labels and pages the result pages cite. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2202.02384), every other right reserved.

For a primitive set A (no member divides another) let f(A) be the sum of 1/(a log a). Theorem 1.2 (p. 3) proves Erdos's conjecture that f(A) <= f(P) = 1.6366... for every primitive set A, improving the previous bounds f(A) < 1.84 of Erdos-Zhang and f(A) < e^gamma = 1.781... of Lichtman-Pomerance. Theorem 1.3 (p. 3) shows every odd prime p is Erdos strong, that is f(A_p) <= f(p) for every primitive A, where A_p is the set of elements of A with least prime factor p; the paper leaves open whether p = 2 is Erdos strong, and completes Theorem 1.2 for sets omitting 2 by the direct bound f(A) < 1.60 of Theorem 4.4 (p. 12). The method refines the Lichtman-Pomerance argument using Mertens' product theorem, the key new input being that a primitive set cannot contain too many elements a whose largest prime factor P(a) is only slightly less than a (Proposition 3.3, p. 9), which gains a factor pi/4 on the contribution of each non-prime element; since e^gamma pi/4 < f(P), the sum is maximized by primes. Theorem 1.5 (p. 3) applies the same method to the 1968 Erdos-Sarkozy-Szemeredi conjecture, proving that the limit as x tends to infinity of the supremum of f(A) over primitive A in [x, infinity) is at most e^gamma pi/4, approximately 1.399, where the conjectured bound is 1. Through L-multiples, the multiples ba of a whose cofactor b has every prime factor at least P(a), the paper also refines the Davenport-Erdos theorem on infinite divisibility chains to L-divisibility chains (Theorem 1.8, p. 5) and extends the 1966 Erdos-Sarkozy-Szemeredi growth bound for such chains under positive upper log log density (Theorem 1.10, p. 5).

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv v4; no proof is checked step by step.

Source: https://arxiv.org/abs/2202.02384.

Bears on.

• #164: Theorem 1.2 proves that the sum of 1/(n log n) over a primitive set is at most its value over the primes, which is the problem's question.
• #1196: Theorem 1.5 bounds the sum over primitive sets in [x, infinity) by e^gamma pi/4 + o(1), which is about 1.399, where the problem asks for 1 + o(1); it does not answer the question.
• #1217: Theorem 1.10 applies to the problem's sets, since positive lower logarithmic density implies positive upper log log density, and gives an infinite L-divisibility chain, in particular a divisibility chain, whose count up to y over log log y has upper limit at least the upper log log density divided by e^gamma, where the problem asks for the density itself; it does not answer the question. The paper records (p. 6) the Erdos-Sarkozy-Szemeredi conjecture that the constant can be improved to the density itself, of which the problem is the case of positive lower logarithmic density.

Results.

• Theorem 1.2 (p. 3): for any primitive set A, f(A) <= f(P).
• Theorem 1.3 (p. 3): for any primitive set A and any prime p > 2, f(A_p) <= f(p).
• Theorem 1.5 (p. 3): the limit as x tends to infinity of the supremum of f(A) over primitive A contained in [x, infinity) is at most e^gamma pi/4, approximately 1.399.
• Theorem 1.8 (p. 5): a set of positive upper log density contains an infinite L-divisibility chain.
• Theorem 1.10 (p. 5): a set A with positive upper log log density contains an infinite L-divisibility chain D with limsup over y of the count of D up to y divided by log log y at least the upper log log density of A divided by e^gamma.

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