Sum of a geometric series over natural density zero indices

Question

It is well known that the geometric power series $% %TCIMACRO{\dsum \limits_{k=0}^{\infty}}% %BeginExpansion {\displaystyle \sum \limits_{k=0}^{\infty}} %EndExpansion x^{k}$ is convergent to $\dfrac{1}{1-x}$ for any $\left \vert x\right \vert <1 $. Is there any infinite set of density zero over which the sum of the corresponding series can be still evaluated? For example, can we find the sum of the series $% %TCIMACRO{\dsum \limits_{k\text{ is prime}}}% %BeginExpansion {\displaystyle \sum \limits_{k\text{ is prime}}} %EndExpansion x^{k}$ for $0<x<1$?

Answer 1

It fully depends on the zero density set you are looking at. Certainly all series converge but it depends on what is the representation of that set. For instance, if the set is $\{\}$ the answer is 0, if the set is the powers of 2 then the sum is $$\sum_{k=0}^\infty x^{2^k}$$ and you could very fastly describe a set of zero density such that the sum cannot be written in closed form, such as Mersenne's primes, numbers of the form $\lfloor \pi n+e^n\rfloor$ and whatnot.

Answer 2

$\sum_{n=0}^{\infty}x^{n^2}$ is a Theta function.