Geodesic
Certain classes of power series that cannot be analytically continued across their circle of convergence
Izvestiya. Mathematics , Tome 61 (1997) no. 4, pp. 795-812

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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 $0\gamma1$; 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 $|z|=1$. 
@article{IM2_1997_61_4_a5,
     author = {A. I. Pavlov},
     title = {Certain classes of power series that cannot be analytically continued across their circle of convergence},
     journal = {Izvestiya. Mathematics },
     pages = {795--812},
     publisher = {mathdoc},
     volume = {61},
     number = {4},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_4_a5/}
}
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A. I. Pavlov. Certain classes of power series that cannot be analytically continued across their circle of convergence. Izvestiya. Mathematics , Tome 61 (1997) no. 4, pp. 795-812. http://geodesic.mathdoc.fr/item/IM2_1997_61_4_a5/