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Simpson, Reginald M.

University of British Columbia

For any cyclic group ℤ, this work studies the number of copies that appear in the invariant factor decomposition of the multiplicative group (ℤ/mℤ)⁢ˣ, denoted ℑ(m; d). The work proves the existence of a sequence of singleton and doubleton sets {fᵢ}ᵢ∞₌₁, a “universal profile” such that the natural density of numbers m where ℑ(m; d) > 0 is 1 for d in the singleton sets, 1/2 for d in the doubleton sets, and 0 for all d not found in the sets. The “universal profile” of sets is fully characterized, and can be explicitly computed based on the description given in this work, and has the property that for d₁ ∈ f ᵢ and d₂ ∈ fⱼ with i < j, d₁ divides d₂. For all d ∈ ℕ, we further determine Erdős-Kac type laws for the limiting distributions of ℑ(m; d) under appropriate scaling. These laws do not always follow the standard Erdős-Kac pattern of having normal limiting distributions, and mathematical descriptions of these non-normal distributions is given. Asymptotic formulas for the expectations of ℑ(m; d) are determined using these laws.

Text

2025-05-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/91229

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/