Cotype zeta function for subrings of ${\mathbb Z}[t]/(t^3)$

The cotype zeta function of $\mathbb{Z}^d$

We give an asymptotic formula for the number of sublattices $Λ\subseteq \mathbb{Z}^d$ of index at most $X$ for which $\mathbb{Z}^d/Λ$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to recent work of Stanley and Wang on Smith Normal Forms of random integral matrices and discuss connections to the Cohen-Lenstra heuristics. Our arguments are based on Petrogradsky's formulas for the cotype zeta function of $\mathbb{Z}^d$, a multivariable generalization of the subgroup growth zeta function of $\mathbb{Z}^d$.

Counting Subrings of ${\mathbb Z}^n$ of finite index

In this article we investigate the number of subrings of $\Z^d$ using subring zeta functions and $p$-adic integration.

计数 $\mathfrak{o}^n$ 的子代数

Counting subalgebras of $\mathfrak{o}^n$

本文提出研究 $\mathfrak{o}^n$ 子代数类型ζ函数的方法，通过有限求和和积分计算，确定有限指数子代数的数量及不可约子环的计数。

Is the number of subrings of index $p^e$ in $\mathbb{Z}^n$ polynomial in $p$?

It is well-known that for each fixed $n$ and $e$, the number of subgroups of index $p^e$ in $\mathbb{Z}^n$ is a polynomial in $p$. Is this true for \emph{subrings} in $\mathbb{Z}^n$ of index $p^e$? Let $f_n(k)$ denote the number of subrings of index $k$ in $\mathbb{Z}^n$. We can define the subring zeta function over $\mathbb{Z}^n$ to be $ζ_{\mathbb{Z}^n}^R(s) = \sum_{k \ge 1} f_n(k)k^{-s}$. Is this zeta function uniform? These two questions are closely related. In this paper, we describe what is known about these questions, and we make progress toward answering them in a couple ways. First, we…

Lower bounds for the number of subrings in $\mathbb{Z}^n$

Let $f_n(k)$ be the number of subrings of index $k$ in $\mathbb{Z}^n$. We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in $\mathbb{Z}^n$, improving upon lower bounds given by Kaplan, Marcinek, and Takloo-Bighash. Further, we prove two new lower bounds for $f_n(p^e)$ when $e \ge n-1$. Using these bounds, we study the divergence of the subring zeta function of $\mathbb{Z}^n$ and its local factors. Lastly, we apply these results to the problem of counting orders in a number field.

$\mathbb{Z}^n$ 的大多数子环具有大余秩

Most subrings of $\mathbb{Z}^n$ have large corank

本文研究具有附加代数结构的子格商，证明当维度 $n \ge 7$ 时 $\mathbb{Z}^n$ 的子环商极少为循环群且通常具有大秩，结合解析数论与组合学及 zeta 函数完成了证明。