Type of an entire function has uncountable zeros

Question

If an entire function has uncountable zeroes then what can you say about the function?

Answer 1

If $f$ is a not identically zero entire function then its set of zeroes has no accumulation points. In particular the number of zeroes of modulus less than or equal to $M$ is finite for all $M>0$. Therefore the number of zeroes is at most countable.