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arXiv · 2012.08006

Bounds for zeros of Collatz polynomials, with necessary and sufficient strictness conditions

Matt Hohertz
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Abstract

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 Enestr\"om-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.

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Matt Hohertz. 2020-12-14. Bounds for zeros of Collatz polynomials, with necessary and sufficient strictness conditions. https://arxiv.org/abs/2012.08006

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