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.

中文翻译：

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Zn 中子环数的下限

让 $F_n(ķ)$是索引k in的子环数${\mathbb{Z}}^n$. 我们表明 Brakenhoff 的结果意味着子环的渐近增长的下界${\mathbb{Z}}^n$，改进了 Kaplan、Marcinek 和 Takloo-Bighash 给出的下限。此外，我们证明了两个新的下界$F_n(p^e)$什么时候$e\ge n-1$. 使用这些界限，我们研究了子环 zeta 函数的发散${\mathbb{Z}}^n$及其局部因素。最后，我们将这些结果应用于在数字字段中计算订单的问题。