Chan 2023 moments gaps between consecutive square free

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Chan, Tsz Ho, On moments of gaps between consecutive square-free numbers. Mosc. J. Comb. Number Theory 12 (2023), no. 4, 287--295. DOI 10.2140/moscow.2023.12.287.

Writing s_1 < s_2 < ... for the squarefree numbers, the paper studies the moment sum of (s_{k+1}-s_k)^gamma over s_{k+1} <= x, which Erdos showed is asymptotic to B(gamma)x for 0 <= gamma <= 2, with B(gamma) built from Mirsky's gap densities alpha(h). Theorem 1 proves the dyadic form of this asymptotic (sum over x/2 < s_{k+1} <= x asymptotic to B(gamma)x/2) for all 0 <= gamma < 3.75, which yields the full asymptotic in that range. This improves the previous best ranges of Hooley (gamma <= 3), Filaseta and Trifonov (43/13), and Huxley (11/3 and then 59/16 = 3.6875). The method refines Huxley's geometric argument: Case 1(a) is treated with a sharper counting result for rational points near a curve, and a fifth-derivative estimate replaces a fourth-derivative one, together with a finer split of the ranges of the gap parameter H. Erdos problem 145 asks for this moment asymptotic for every gamma >= 0; the paper settles the range 0 <= gamma < 3.75 and records that further progress hinges on improving Huxley's Case 2.

The held PDF is the arXiv preprint arXiv:2310.08448v1 (12 October 2023, 10 pp., the only arXiv version; its title spells "squarefree"); page and theorem numbers on this card are those of that copy, Theorem 1 on p. 2. The journal version above, also cited on Problem 145, was not read or compared with it.

Source: https://arxiv.org/abs/2310.08448. The arXiv record (https://arxiv.org/abs/2310.08448, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #145

Results to transcribe.

• Theorem 1 (p. 2): For 0 <= gamma < 3.75, the sum of (s_{k+1}-s_k)^gamma over x/2 < s_{k+1} <= x is asymptotic to B(gamma)x/2, hence the full moment asymptotic holds for that range of gamma.
• Constant B(gamma): B(gamma) = sum over h >= 1 of h^gamma alpha(h), convergent because log alpha(h) <= -(5/4) h log log h + O(h) (Huxley's Lemma 1).