Zeta functions and asymptotics related to subrings in $\mathbb{Z}^n$

We can define a zeta function of a group (or ring) to be the Dirichlet series associated to the sequence that counts the number of subgroups (or subrings) of a given index. The subgroup zeta function over $\mathbb{Z}^n$ is well-understood, as is the asymptotic growth of subgroups in $\mathbb{Z}^n$. Much less is known about the subring zeta function over $\mathbb{Z}^n$ and the asymptotic growth of subrings in $\mathbb{Z}^n$. In this talk, we discuss the progress toward answering this question and we give new lower bounds on the asymptotic growth of subrings in $\mathbb{Z}^n$. We also define a similar zeta function corresponding to subrings of corank at most k in $\mathbb{Z}^n$. While the proportion of subgroups in $\mathbb{Z}^n$ of corank $k$ is positive for each $k$, we show this is not the case for subrings in $\mathbb{Z}^n$ of corank $k$ when $n$ is sufficiently larger than $k$. Lastly, we make connections to orders in number fields. Part of this work is joint with Nathan Kaplan.