The following appears as the second-to-last problem of Stewart's *Complex Analysis*:

Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$.

This problem intimidated me when I first saw it as an undergrad, as the series under the root isn't the expansion of any function I know (then or now). That renders the Riemann surface more subtle than usual, since any info about poles and zeroes must be found from this series alone. For instance, it has a zero at $z=0$ and therefore a branch point. But there should be other zeroes — indeed, *infinitely many zeroes*— since the number associated with any truncation of the series grows as $2^N$!. This suggests that the Riemann surface of this function must be exotic.

I'll give a sketch of an argument below, taking the pedestrian approach of hunting poles and zeroes of the function. However, I'd be glad to see an elegant and advanced perspective of the problem, particularly if it uses some ideas which I wouldn't have encountered as an undergraduate. The more food for thought, the better!

EDIT: As noted by O.L.'s answer below, the series above is the canonical example of a lacunary function with the unit circle as natural boundary. What that leaves open is the nature of the resulting Riemann surface, which would seem to hinge upon the quantity of the function's zeros within the unit circle. David Speyer has given an argument in his answer that this number is at least 10. **Is there a more definitive result on the zeroes of this lacuuary function?**