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.