Higher-power divisibility in a floor function set
Issued Date
2026-02-01
Resource Type
ISSN
15131874
Scopus ID
2-s2.0-105039245480
Journal Title
Scienceasia
Volume
52
Issue
1
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SCOPUS
Bibliographic Citation
Scienceasia Vol.52 No.1 (2026)
Suggested Citation
Panraksa C. Higher-power divisibility in a floor function set. Scienceasia Vol.52 No.1 (2026). doi:10.2306/scienceasia1513-1874.2026.012 Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/116849
Title
Higher-power divisibility in a floor function set
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Abstract
Let (Formula presented) and write 1<inf>S(x)</inf> for its indicator. For fixed k⩾3 and a multiplicative function g, put (Formula presented). We study (Formula presented), and obtain explicit bounds for the error term E<inf>k,g</inf>(x) across three natural classes (Types I–III) of multiplicative g. Our arguments use the distribution of S(x) in arithmetic progressions due to Yu and Wu, which yields (Formula presented) uniformly for (Formula presented). Consequently, all unconditional results here are uniform in this proven range; extensions to m⩽x follow conditionally under a divisible–subset alignment assumption. The case k=2 is recovered as a special instance; for k⩾3 we isolate the new features arising from higher-power divisibility, including a small/large-d decomposition tuned to the Yu–Wu range and explicit k-dependent exponents in E<inf>k,g</inf>(x). We also include short worked examples for g≡1, g=μ, and g=μ<inf>2</inf>.