An entire function whose every derivative-subsequence has dense zero set: does one exist? (Erdős #906)
Statement
Is there an entire non-zero function $f:\mathbb{C}\to\mathbb{C}$ such that, for every infinite sequence $n_1<n_2<\cdots$ of positive integers, the set $$\{z\in\mathbb{C}: f^{(n_k)}(z)=0 \text{ for some } k\ge 1\}$$ is everywhere dense in $\mathbb{C}$? Here $f^{(m)}$ denotes the $m$th derivative; the intended $f$ is transcendental, since for a polynomial the high derivatives vanish identically and the question is trivial.
Acceptance. FULLY RESOLVES: either construct an explicit entire non-zero (transcendental) function $f$ and give a complete proof (machine-checkable in Lean/Coq preferred, otherwise a full written proof) that for every infinite sequence $n_1<n_2<\cdots$ the union $\{z: f^{(n_k)}(z)=0 \text{ for some } k\}$ is dense in $\mathbb{C}$; or prove that no such entire function exists. ADVANCES: prove existence of an $f$ satisfying a weaker but still nontrivial variant (e.g. density only along the real axis, or density for every subsequence of positive lower density), with proof; or establish structural constraints that any such $f$ must satisfy (order of growth, distribution of zeros of derivatives); or reconstruct and rigorously verify the affirmative proof Erdős attributed to the early-1970s literature, with a precise, checkable citation. Deliver the construction with its proof, the impossibility proof, or the verified rediscovered proof.
Background
Posed by Erdős [Er56d, Er82e p.72]. The condition demands that the zeros of the successive derivatives $f^{(m)}$ be so pervasively spread that no matter which infinite subset of derivative orders one selects, the union of the corresponding zero sets is dense in the plane. A subtle wrinkle in the history: in [Er82e] Erdős writes that this problem was solved in the affirmative 'more than ten years ago' but gives no reference and does not name the solver; from context he appears to attribute it to Barth and Schneider [BaSc72], yet that paper contains no such result. Bloom therefore lists the problem as still open. A commenter (Tang) points out that the problem is trivial for polynomials, so $f$ must be taken transcendental. Listed as open on erdosproblems.com/906 (fetched 2026-07-21, status 'open'). NOTE FOR THE REVIEWER: the 'open' status is mildly contested — Erdős himself believed an affirmative solution existed, but the attributed source does not support it and no other reference is known; a solver could equally supply a fresh proof or recover the lost one. Attacker's tool: constructive complex analysis — build a transcendental entire function (via Weierstrass/canonical products or gap series) with prescribed control over the zeros of all its derivatives (Pólya–Wiman theory and the geometry of zeros of successive derivatives), or a Baire-category / normal-families existence argument; alternatively track down and verify the affirmative proof Erdős alluded to.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #906 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.