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I've started reading Shakarchi's Complex Analysis, and I thought about something interesting.

If I haven't mistaken, for any subsequence $A\subset \mathbb{Z}^+$, $\sum_{n\in A} z^n$ has radius of convergence $1/(\limsup |a_n|^{1/n})=1$.

So we can take $f(z)=\sum_p z^p$ for prime $p$, which has radius of convergence 1. This looks like a really fascinating function to me. Does anyone know a prior study done on this subject?

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    A [related question](http://math.stackexchange.com/questions/55626/asymptotic-behavior-of-sum-i0-xp-i-as-x-to-1) may be of interest.2012-10-15

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