A proof that entire function of order which is a fraction have infinitely many zeros

Question

I understand that it is the case, I can give examples like $cos(a\sqrt{z})$ or $\Xi(\sqrt{z})$ but I would like to see a strict proof and I don't know how to create one myself.

Answer 1

By Picard's great theorem, a trascendental entire function (i.e. not a polynomial) takes on any neighborhood of infinity all complex values an infinite number of times with at most one exception. An entire function of non integer order is not a polynomial, so that Piccard's theorem applies. But more is true in this case: there is no exception, that is, an entire function whose order is not an integer takes all complex values an infinite number of times. The proof is not straightforward, and is based on the relation between the order of the function and the order of convergence of its zeroes.