I was thinking about the following problem:
Let $S=\{0\}\cup \{ \frac{1}{4n+1}:n=1,2,3,4,\dots\}$ Then what is the total number of analytic function which vanish only on $S$?
I was trying to use the fact that zeros of analytic functions are isolated. But I could not progress further. Am I going in the right direction? Please help. Thanks in advance for your time.