Holomorphic motion, rational approximation and an equivalent formulation of the Riemann Hypothesis

Abstract

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform limits of polynomials (respectively rational functions, having no poles on $K$) and $A(K)$ is the family of continuous functions on $K,$ which are holomorphic on the interior of $K.$ In the class of compact sets, the property of being a set of polynomial approximation is easily seen to be invariant under holomorphic motion. We show that this is no longer the case for rational approximation. Secondly, we show that the Riemann Hypothesis holds if and only if a certain map is a holomorphic motion.