Functional Equations Associated to Collatz-Type Maps on Integer Rings of Algebraic Number Fields

TL;DR

This paper extends the functional equation approach to analyze Collatz-type maps from integers to rings of integers in algebraic number fields, broadening the scope of previous reformulations of the Collatz Conjecture.

Contribution

It generalizes the functional equation method to Collatz-type maps on algebraic number fields, expanding the analytical framework beyond integers.

Findings

Generalized Collatz maps on algebraic number fields analyzed

Extended the functional equation approach to new algebraic structures

Provided a foundation for future research on Collatz conjecture in broader settings

Abstract

In 1995, Meinardus & Berg presented a reformulation of the Collatz Conjecture in terms of a functional equation in a single complex variable over the open unit disk. This paper generalizes that method to deal with not only a large class of Collatz-type maps defined on the integers, but further generalizations thereof to Collatz-type maps on the rings of integers on an arbitrary algebraic number field.