Analytic Observables Along Recursive Complex Root Orbits

This project studies the behavior of complex numbers generated through repeated root extraction with a fixed branch choice. It develops the resulting dynamics, describes the full structure of the possible root choices, and introduces a natural way to encode their recursive phase information. The paper then examines how analytic functions behave along these recursively generated complex orbits, with the Riemann zeta function used as an example. The goal is to provide a simple mathematical framework for studying recursive complex root processes and the structures that arise from them.