Abstract:
We define, in number-theoretical terms, the class $\{M\}$ of sets of natural numbers having the properties:
1) the asymptotic density $\gamma$ of a set $M$ satisfies the inequality $0<\gamma<1$;
2) if $G(z)$ is an entire function with non-negative Taylor coefficients and not growing too fast at infinity, then the power series $\sum_{m\in M}G(m)z^m$, having radius of convergence 1, cannot be analytically continued into the domain $|z|>1$ across any arc on the circle $|z|=1$.
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