Zero-free polynomial approximation on a chain of Jordan domains

Abstract

On a compact subset of the plane with connected complement, is it possible to uniformly approximate a continuous function, holomorphic and non-vanishing on the interior, with polynomials non-vanishing on the entire compact set? In this brief note, we recall the surprising connection between this question and the Riemann hypothesis and proceed to provide an affirmative answer for a "chain" of Jordan domains.