Cofinite Zeros of High Derivatives

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 Erd\H{o}s Problem~906. Thus every fixed disk is zero-free for only finitely many successive derivatives. The function satisfies $\mathrm{\mid }f(z)\mathrm{\mid }\le \sqrt{2}\exp (\mathrm{\mid }z{\mathrm{\mid }}^2)$ and is a counterexample to a theorem of Boas and Reddy as printed.