Counting Finite Index Subrings of $\mathbb{Z}^n$
  We count subrings of small index of $\mathbb{Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of subrings of index $k$. For any $n$, we give a formula for this quantity for all integers $k$ that are not divisible by a 9th power of a prime, extending a result of Liu. 
Stanislav Atanasov, Nathan Kaplan, Benjamin Krakoff, Julia Menzel
math.CO math.NT