Cofinite Zeros of High Derivatives

Eric Hou

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arXiv · 2607.20816

Cofinite Zeros of High Derivatives

Eric Hou

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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 Erd\H{o}s Problem~906. Thus every fixed disk is zero-free for only finitely many successive derivatives. The function satisfies $|f(z)|\leq\sqrt2\exp(|z|^2)$ and is a counterexample to a theorem of Boas and Reddy as printed.

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Eric Hou. 2026-07-23. Cofinite Zeros of High Derivatives. https://arxiv.org/abs/2607.20816

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