analysis / Analysis, Entire Functions
Is there an entire non-zero function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\cdots$, the set $\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\}$ is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.
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Append-only history
Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference
Research memory
Is there an entire non-zero function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\cdots$, the set $\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\}$ is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.
Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference
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