Problem Statement
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?
Categories:
Analysis Iterated Functions
Progress
Erdős [Er82e] writes that this was solved in the affirmative 'more than ten years ago', but gives no reference or indication who solved it. From context he seems to attribute this to Barth and Schneider [BaSc72], but this paper contains no such result.Tang points out that the problem is trivial if we take $f$ to be a polynomial, so presumably it is intended the function $f$ is transcendental.
Source: erdosproblems.com/906 | Last verified: January 19, 2026