Higher-power divisibility in a floor function set

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>.