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

TL;DR

This paper refines bounds on the roots of Collatz polynomials using the Enestr"om-Kakeya Theorem, establishing strictness conditions and confirming that roots with modulus 2 are rare among positive integers.

Contribution

It introduces tighter bounds for Collatz polynomial roots and provides necessary and sufficient conditions for these bounds to be sharp, advancing understanding of their root distribution.

Findings

Roots on the circle |z|=2 are rare among Collatz polynomials

New bounds are sharper and more precise than previous results

Conditions for bounds to be tight are explicitly characterized

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.