Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

Abstract

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as $n\to\infty$. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.