Asymptotic Results on Subrings in Z^n of Corank at most k

Abstract: The field of subgroup growth has become an active field in number theory. In this talk, we consider subring growth in the ring $\mathbb{Z}^n$. While the exact growth rate of the number of subgroups in $\mathbb{Z}^n$ is known, there is not even a conjecture about what the growth rate of the number of subrings in $\mathbb{Z}^n$ should be, though several authors have produced upper and lower bounds. In this talk, we consider subrings of corank $k$ and define a zeta function that carries information about the number of subrings in $\mathbb{Z}^n$ of a fixed index and corank at most $k$. We express the corank zeta functions for $\mathbb{Z}^n$ when $n \le 4$ in terms of simpler Euler products. We also show that most subrings in $\mathbb{Z}^n$ have large corank, and we compare this to recent results about subgroups in $\mathbb{Z}^n$ with fixed corank. This is joint work with Nathan Kaplan.

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