Interestingly, there's not only theorems but also conjectures in this list.
Interesting bit:
> ITEM 125 (Polya):
> CONJECTURE: If a function has a power series with integer coefficients and radius of convergence 1, then either the function is rational or the unit circle is a natural boundary.
> Reference: Polya, Mathematics and Plausible Reasoning, volume 2, page 46.
Has this conjecture been proved or disproved by now?