Counting Subrings of $\Bbb Z^n$

How many subgroups of $\Bbb Z^n$ have index at most $X$?

How many of these subgroups are also subrings?

We can give an asymptotic answer to the first question by computing the subgroup zeta function of $\Bbb Z^n$. For the second question, we only know an asymptotic answer for small $n$ because the subring zeta function of $\Bbb Z^n$ is much harder to compute. It is not difficult to show that it is enough to understand the number of subrings of prime power index.

Let $f_ n(p^e)$ be the number of subrings of $\Bbb Z^n$ with index $p^ e$. When $n$ and $e$ are fixed, how does $f_ n(p^e)$ vary as a function of $p$?

We will discuss the quotient $\Bbb Z^n/L$, where $L$ is a ‘random’ subgroup or subring of $\Bbb Z^n$. We will also see connections to counting orders in number fields.

Nathan Kaplan