  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.
