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

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.