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

Abstract

In a previous paper, we introduced the Collatz polynomials $P_N\left(z\right)$, 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 $P_N\left(z\right)$ and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $\{z\in \mathbb{C}:\left|z\right|=2\}$ are rare: the set of N such that $P_N\left(z\right)$ has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study.

Keywords:

• Polynomial zeros
• Collatz conjecture
• generating functions

AMS Subject Classification:

• 30C15

Acknowledgments

I would like to thank Harold Boas for his suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 cf. item 9 of Section 5