Certain classes of power series that cannot be analytically continued across their circle of convergence

Abstract

We define, in number-theoretical terms, the class of sets of natural numbers having the properties: 1) the asymptotic density of a set satisfies the inequality ; 2) if is an entire function with non-negative Taylor coefficients and not growing too fast at infinity, then the power series , having radius of convergence 1, cannot be analytically continued into the domain across any arc on the circle .