Lower bounds for the number of subrings in ℤⁿ

2020/10/18 by Kelly Isham, Isham, Kelly
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2010.09123

openalex publication_date 2020/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f_n(k) be the number of subrings of index k in ℤⁿ. We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in ℤⁿ, improving upon lower bounds given by Kaplan, Marcinek, and Takloo-Bighash. Further, we prove two new lower bounds for f_n(p^e) when e ≥ n-1. Using these bounds, we study the divergence of the subring zeta function of ℤⁿ and its local factors. Lastly, we apply these results to the problem of counting orders in a number field.

Related

• Counting Subrings of \mathbb Zⁿ of finite index
• Is the number of subrings of index p^e in ℤⁿ polynomial in p?
• On the number of subrings of ℤⁿ of prime power index
• Subrings of \mathbb Z[t]/(t⁴)
• Counting subrings of \mathbb Zⁿ of non-zero co-rank