Lower bounds for the number of subrings in ${\mathbb{Z}}^n$

Abstract

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

arXiv commentaccepted to JNT, minor edits based on the referee's feedback, typos corrected, 22 pages