Problem 906

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claims/: The 4 claim pages of Problem 906, one per claimant's result; the problem's standing derives from them.

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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?

Statement (corrected). Is there an entire transcendental 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?

Notes. The site's wording is trivially true: every non-zero polynomial, the constant $1$ among them, is an entire non-zero function whose derivatives of order above its degree vanish identically, so for every sequence $n_1<n_2<\cdots$ the set contains the whole plane. The failure covers the whole class of polynomials, so it is a failure of setting, not of range. Tang pointed it out, as the site's commentary records. The change replaces "non-zero" by "transcendental"; a transcendental entire function is non-zero, and nothing else changes. The evidence is the site's own commentary, which records the polynomial failure and gives the intended form, $f$ transcendental, while the site keeps the label OPEN, which only that form fits; the formal-conjectures statement file, which counts with the site, states the same form. The defect is already in the poser's text: Erdős's question (i) in [Er82e, p. 72, §IV.1] asks for an entire function $f(z)$ with the roots of the $f^{(n_i)}$ everywhere dense, with no condition that excludes polynomials, and the site's "non-zero" excludes only the zero function. The correction does not rest on Erdős's Hungarian paper (Some remarks on a paper of Kővári, Mat. Lapok 7 (1956), 214--217), where [Er82e] says the question was raised, or on [BaSc72]. The form follows from the site's commentary and label, not from the results that settle it. The polynomial observation answers the site's wording only and counts for nothing; no claim page records it, since no one presented it as settling the problem. The page's standing judges the corrected Statement.

Status. The site labels the problem OPEN (page last edited 1 October 2025), a label that describes the corrected Statement. Four pending full claims, each with declared AI assistance, construct such a transcendental function: Almeida's planar Gaussian series and Chojecki's Gaussian series, both posted on the site's discussion thread on 25 April 2026, and, on the site's proof-claims tab, Hou's random Fock series of 21 July 2026 and He's sparse Fock series of 17 September 2026, the last of which does not claim to be the first solution. A 1973 theorem of Boas and Reddy, as printed, excludes three of the four constructions, and the conflict is unresolved. The site has accepted none of them, this corpus has built neither of the two Lean developments, and the derived standing is claimed.

Source. erdosproblems.com/906, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #906, https://www.erdosproblems.com/906.

References.

• [BaSc72] Barth, K. F. and Schneider, W. J., On a problem of Erdős concerning the zeros of the derivatives of an entire function. Proc. Amer. Math. Soc. (1972), 229-232.
• [Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59-79.

Formalization. Statement in formal-conjectures, which at the revision linked asks for a transcendental entire function with the density property and tags the question open.

Current assessment

The question (page last edited 1 October 2025). The corrected Statement above: a transcendental entire $f$ such that for every infinite sequence $n_1<n_2<\cdots$ the zeros of the derivatives $f^{(n_k)}$, taken together, are dense in $\mathbb{C}$. The site's wording, which also admits polynomials, is answered yes by any non-zero polynomial, as the Notes record. The site's label is OPEN.

History. Erdős [Er82e] writes that the existence of the function of his question (i), the problem here, and of the function of his question (ii) had been proved more than ten years before; on p. 72 (section IV.1) the sentence carries footnote (1), which cites Barth and Schneider, Proc. Amer. Math. Soc. 32 (1972), 229--232 [BaSc72], and three other papers of theirs (J. reine angew. Math. 234 (1969); J. London Math. Soc. (2) 2 (1970); J. London Math. Soc. (2) 4 (1972)). The site's commentary says that Erdős gives no reference and that the 1972 paper contains no such result; that report is the site's. No published solution is recorded on the site or on this wiki.

Pending claims. Four full claims each construct a transcendental entire function whose high derivatives have a zero in every fixed nonempty open set, which is equivalent to the question's density condition and so answers the corrected Statement yes. Two were posted on the site's discussion thread on 25 April 2026, each with a dated manuscript. Almeida's claim, whose manuscript says it was prepared with AI assistance and whose document metadata names ChatGPT, uses the planar Gaussian entire function $\sum_k\xi_kz^k/\sqrt{k!}$ with independent standard complex Gaussian coefficients, an expected zero count of order $\sqrt n$ from the Edelman-Kostlan formula, an Offord-type bound on the probability of a zero-free disk, and the Borel-Cantelli lemma over rational disks. Chojecki's claim, a note presented as done with GPT-5.5 Pro, uses the Gaussian series $\sum_k\xi_kz^k/(k!)^{1-\beta}$ with $1/2<\beta<1$; its claimant later confirmed on the thread that it is a full solution and that Almeida posted first. Hou's claim, filed on the proof-claims tab on 21 July 2026 with assistance from the AI system named as ChatGPT 5.6 Sol, uses a random Fock series with independent coefficients uniform on the unit disk, a hole-probability estimate through Jensen's formula, and a growth bound $\lvert f(z)\rvert\le\sqrt2\exp(\lvert z\rvert^2)$, with a Lean development whose self-reported axiom report is the three standard axioms; its August 2026 revision credits the two earlier proposals. He's claim, filed on the tab on 17 September 2026 with assistance from the systems named as GPT-5.6 Sol, GPT-6 Astra and Aristotle, uses the explicit sparse Fock series with exponents $\lfloor j^{3/2}\rfloor$, locating the zeros of $F^{(n)}$ within $Cn^{-1/6}$ of every point of a fixed annulus, which strengthens the density condition quantitatively, with a Lean development, and does not claim to be the first solution. The site has accepted none of them, and this corpus has built and audited neither Lean development, so all four stay claimed and the derived standing is claimed, with the claim value proved, since all four assert the affirmative answer.

A conflicting theorem. Theorem 1 of Boas and Reddy (Bull. Amer. Math. Soc. 79 (1973), 64-65; expanded in J. Math. Anal. Appl. 42 (1973), 466-473), as Hou's manuscript reports it, states that a transcendental entire function of order at most 2 and finite type has arbitrarily large disks on each of which infinitely many of its derivatives have no zero, so no such function has the property asked for. Hou's and He's functions satisfy $\lvert f(z)\rvert\le\sqrt2\exp(\lvert z\rvert^2)$, and Almeida's Gaussian series has order 2 and type $1/2$ almost surely. So each of these three claims, if correct, refutes that theorem as printed, as Hou's manuscript states. Chojecki's functions have order $1/(1-\beta)>2$ and lie outside it. The conflict is unresolved.

Search scope. The site's problem page and proof-claims tab as cached on 2026-10-06, and the site's discussion thread of the problem. No refereed work beyond the references was found.

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