论文标题
$ \ Mathbb {z}^n $中子环数的下限
Lower bounds for the number of subrings in $\mathbb{Z}^n$
论文作者
论文摘要
令$ f_n（k）$为$ \ mathbb {z}^n $的索引$ k $的子环数。我们表明，布雷克霍夫（Brakenhoff）的结果意味着在$ \ mathbb {z}^n $中的子环的渐近生长中有一个下限，在Kaplan，Marcinek和Takloo-Bighash给出的下限上有所改善。此外，当$ e \ ge n-1 $时，我们证明了两个新的下限（p^e）$。使用这些边界，我们研究了$ \ Mathbb {z}^n $及其局部因素的子Zeta函数的差异。最后，我们将这些结果应用于数字字段中计数订单的问题。
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.
