论文标题
Collatz多项式的零的界限，具有必要和充分的严格条件
Bounds for zeros of Collatz polynomials, with necessary and sufficient strictness conditions
论文作者
论文摘要
在上一篇论文中，我们介绍了Collatz多项式$ P_N（Z）$，其系数是正整数$ n $的Collatz序列的术语。我们在本文中的工作扩展了我们以前的结果，使用Eneström-kakeya定理来收紧$ p_n（z）$的根源的旧边界，并提供这些新界限的精确条件。特别是，我们确认了一个实验结果，即圆上的零$ \ {z \ in \ mathbb {c}：| z | | = 2 \} $很少见：$ n $的集合，使$ p_n（z）$具有模量2的根在自然数中很少。我们解决了一些问题以进行进一步研究。
In a previous paper, we introduced the Collatz polynomials $P_N(z)$, whose coefficients are the terms of the Collatz sequence of the positive integer $N$. Our work in this paper expands on our previous results, using the Eneström-Kakeya Theorem to tighten our old bounds of the roots of $P_N(z)$ and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $\{z\in\mathbb{C}: |z| = 2\}$ are rare: the set of $N$ such that $P_N(z)$ has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study.
