Cofinite Zeros of High Derivatives

Abstract

We give a bounded-coefficient probabilistic construction of a transcendental entire function $f$ of order two such that every nonempty open subset of the complex plane contains a zero of $f^{(n)}$ for all sufficiently large $n$. This gives an affirmative answer to the transcendental form of Erdős Problem~906. Thus every fixed disk is zero-free for only finitely many successive derivatives. The function satisfies $|f(z)|\le \sqrt{2}\exp (|z{|}^2)$ and is a counterexample to a theorem of Boas and Reddy as printed.

10 Pages