let $k$ be a number field of degree $n$ and $\mathcal{o}_k$ be its ring of integers. an order in $\mathcal{o}_k$ is a finite index subring that contains the identity. a major open question in arithmetic statistics asks for the asymptotic growth of orders in $k$. in this talk, we will give the best known lower bound for this asymptotic growth. the main strategy is to relate orders in $\mathcal{o}_k$ to subrings in $\mathbb{z}^n$ via zeta functions. along the way, we will give lower bounds for the asymptotic growth of subrings in $\mathbb{z}^n$ and for the number of index $p^e$ subrings in $\mathbb{z}^n$. we will also discuss analytic properties of these zeta functions.