Volume 49 Number 3

Higher-power divisibility in a floor function set

Chatchawan Panraksa^†

 
ABSTRACT:     LetS(x)={ x/n :1 n x}andwrite1S(x) foritsindicator. Forfixed k 3andamultiplicativefunction g, put hk(n) = dk|n g(d). We study 1S(x)(n)hk(n) = 2x1/2 n x g(d) dk + Ek,g(x), d 1 and obtain explicit bounds for the error term Ek,g(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 |Sm(x)| = 2x1/2/m+O(x/m)1/3logx uniformly for 1 m x1/4(logx)?3/2. 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 Ek,g(x). We also include short worked examples for g ? 1, g = ?, and g = ?2.

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* Corresponding author, E-mail: chatchawan.pan@mahidol.ac.th

Received 4 Oct 2025, Accepted 1 Feb 2026