The universal profile of the invariant factors of (\mathbb Z/n\mathbb Z)^×

2025/04/15 by Martin, Greg, Simpson, Reginald M.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.11452

Abstract

The structure of the multiplicative group M_n = (\mathbb Z/n\mathbb Z)^× encodes a great deal of arithmetic information about the integer n (examples include ϕ(n), the Carmichael function λ(n), and the number ω(n) of distinct prime factors of n). We examine the invariant factor structure of M_n for typical integers n, that is, the decomposition M_n ≅ \mathbb Z/d₁\mathbb Z × \mathbb Z/d₂\mathbb Z × ⋯ × \mathbb Z/d_k\mathbb Z where d₁| d₂|⋯| d_k. We show that almost all integers have asymptotically the same invariant factors for all but the largest factors; for example, asymptotically 1/2 of the invariant factors equal \mathbb Z/2\mathbb Z, asymptotically 1/4 of them equal \mathbb Z/12\mathbb Z, asymptotically 1/12 of them equal \mathbb Z/120\mathbb Z, and so on. Furthermore, for positive integers k, we establish a theorem of Erdős-Kac type for the number of invariant factors of M_n that equal \mathbb Z/k\mathbb Z, except that the distribution is not a normal distribution but rather a skew-normal or related distribution.

Related

• The smallest invariant factor of the multiplicative group
• The distribution of the number of subgroups of the multiplicative group
• The least primary factor of the multiplicative group
• Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem
• An Erdős-Kac theorem for integers with dense divisors