Dai 2006 sequences bounded l c m each

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theorem: An explicit remainder for the largest set of positive integers whose pairwise least common multiples are at most x.

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Dai, Li-Xia and Chen, Yong-Gao, Sequences with bounded l.c.m. of each pair of terms. {II}. Acta Arith. (2006), 315-326.

The paper is Acta Arithmetica 124 (2006), no. 4, 315--326, DOI 10.4064/aa124-4-2 (Crossref record read). The retained folder-name PDF is the publisher's 12-page file; printed p. $n$ is PDF p. $n-314$, and the text layer is clean. Read status: claims checked for the Theorem, the Remark and the Conjecture, read clause by clause on the page images of pp. 315--316; the proof (pp. 316--326) was not read beyond the statement of Lemma 1. The file's text layer carries no copyright or license line; the publisher's record labels the PDF download "Pobierz zgodnie z CC-BY", which the English site renders "Free download under CC-BY license", a Creative Commons Attribution license with no version or URL named (https://www.impan.pl/get/doi/10.4064/aa124-4-2, read 2026-10-02); the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Let A_x be a largest set of positive integers in which the least common multiple of every pair of terms is at most x. Extending Chen's asymptotic |A_x| = sqrt(9x/8) + o(sqrt x), the paper's Theorem makes the remainder explicit: |A_x| = sqrt(9x/8) + R(x) with -2 <= R(x) <= 45 sqrt(x/log x) log log x for large x, and the authors remark that the constant 45 can be improved. They conjecture R(x) tends to infinity. The proof runs through Brun's pure sieve (Lemma 1) together with Rosser-Schoenfeld estimates for sum 1/p (Lemma 2), applied to the near-extremal structure B_x consisting of the integers up to sqrt(x/2) plus the even integers between sqrt(x/2) and sqrt(2x). This is the sharpest quantitative form cited for Erdos problem 441, which asks for the exact value or asymptotics of |A_x| (Erdos, 1951).

Source: https://doi.org/10.4064/aa124-4-2.

Bears on. #441

Results to transcribe.

• Theorem (pp. 315-316): For large x the maximum size of a set with all pairwise l.c.m.'s at most x satisfies |A_x| = sqrt(9x/8) + R(x) with -2 <= R(x) <= 45 sqrt(x/log x) log log x (the factor log log x outside the square root).
• Conjecture (Sec. 1, p. 316): The remainder R(x) tends to infinity as x tends to infinity.
• Lemma 1 (p. 316): Brun's pure sieve in the form S(A;P,z) = X W(z){1 + theta(lambda e^{1+lambda})^{(A_0A_1/lambda)(log log z+1)}} + theta'(1 + sum_{p<z} omega(p))^{(A_0A_1/lambda)(log log z+1)} with |theta|, |theta'| <= 1, under the lemma's conditions on r_d, omega(p), lambda and sum_{p<z} 1/p; the sieve engine for the theorem.