Cofinite Zeros of High Derivatives

Abstract

We construct a transcendental entire function $f$ for which every nonempty open subset of the complex plane contains a zero of $f^{(n)}$ for all sufficiently large~$n$. Earlier Gaussian proposals by Almeida and Chojecki use expected-zero and Offord-type estimates. Here the coefficients are independent and uniformly distributed on the closed unit disk. A saddle-point estimate, one-coordinate anti-concentration and the mean value property of $\log|f^{(n)}|$ give summable fixed-disk hole probabilities. The selected function has order exactly two and satisfies $|f(z)|\leq\sqrt2\exp(|z|^2)$. It is therefore a counterexample to Boas and Reddy's printed assertion that every transcendental entire function of order at most two and finite type admits an arbitrarily large fixed disk zero-free for infinitely many successive derivatives.

Disclosure

“hanks Xi Chen for detailed suggestions that improved the introduction, theorem formulation, order calculation and circle-average estimate. Declarations Use of AI. In July–August 2026, the author used OpenAI GPT-5.6 Sol and Anthropic Claude Fable 5 to assist with the small-ball estimate and editing, source retrieval, manuscript revision and independent argument auditing. The author checked every”

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2026-08-23T21:12:55.442889Z

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