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

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 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 ∈ 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.