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Claim. The explicit entire function

$$F(z)=\sum_{j\ge1}\frac{z^{\lfloor j^{3/2}\rfloor}}{\sqrt{\lfloor j^{3/2}\rfloor!}}$$

has the property that for all sufficiently large $n$ every point of a fixed annulus lies within $Cn^{-1/6}$ of a zero of $F^{(n)}$, so every nonempty open set contains a zero of $F^{(n)}$ for all large $n$; for every increasing sequence $n_1<n_2<\cdots$ the zeros of the $F^{(n_k)}$ are then dense, the affirmative answer to the corrected Statement of Problem 906, which asks for a transcendental function. As the tab entry describes the argument, near certain crossing radii two neighboring terms of $F^{(n)}$ dominate; their two-term model has equally spaced zeros, and each model zero is shown to correspond to exactly one simple zero of the full derivative nearby, while away from the crossing regions one term dominates and there are no further zeros on a fixed annulus; the crossing radii are spaced on the scale $n^{-1/6}$, which gives the distance bound. The submitter keeps the submission to the exponent $3/2$ matched by the public Lean development, says that broader material in the same repository is not part of the submission, and does not claim that this is the first solution of the problem. The claimant is ZhiJie He, with the claim posted by the user FDmd-233; the entry declares assistance from the AI systems named as GPT-5.6 Sol, GPT-6 Astra and Aristotle, with Astra used in exploration and checking, Aristotle in the Lean formalization, and GPT-5.6 Sol and Codex in checking details and writing. This page's account of the manuscript follows the tab entry's summary and notes.

Submission note. Posted to erdosproblems.com as a proof claim by ZhiJie He (account FDmd-233) on 17 September 2026, giving "GPT-5.6 Sol; GPT-6 Astra; Aristotle" as the AI used:

I use the explicit sparse Fock series

$$> F(z)=\sum_{j\ge1}\frac{z^{\lfloor j^{3/2}\rfloor}}{\sqrt{\lfloor j^{3/2}\rfloor!}}. >$$

For high derivatives, two neighbouring terms dominate near certain crossing radii. Their two-term model has equally spaced zeros, and I prove that each model zero gives exactly one simple zero of the full derivative nearby. Away from these crossing regions, one term dominates, so there are no additional zeros on a fixed annulus. The crossing radii are spaced on the scale $n^{-1/6}$. It follows that every point of a fixed annulus lies within $Cn^{-1/6}$ of a zero for all sufficiently large $n$. This is the part of the argument I find most useful: it immediately implies that every nonempty open set contains a zero of $F^{(n)}$ for every sufficiently large $n$. Notes: I used Astra quite heavily during the exploratory and checking stages of this work, and I used Aristotle to help with the Lean formalization. GPT-5.6 Sol and Codex were also useful for checking details and writing. I have kept this submission deliberately narrow: it only includes the $p=3/2$ result matched by the public Lean development. The broader research material in the repository is not part of this submission. I do not claim that this is the first solution of the problem.

Standing. The claim was filed on the site's proof-claims tab on 17 September 2026 as a full proof, with no comments. The site's label is OPEN with the page last edited 1 October 2025, before the claim, and its commentary does not mention it. No refereed publication and no outside review was found. The earlier claim on Hou's page constructs a different witness, a random Fock series. The function has order 2 and finite type, since $\lvert F(z)\rvert\le\sqrt2\exp(\lvert z\rvert^2)$ by the Cauchy-Schwarz inequality, so it falls under Theorem 1 of Boas and Reddy (1973) as printed; neither the entry nor the manuscript addresses that theorem, and the conflict is recorded on the problem page. The claim stays claimed.

Formalization. The linked folder of the claimant's repository, at the commit pinned in the links, holds a Lean development that the entry says matches the exponent $3/2$ result. This corpus has not built or audited it, so no formalized evidence is listed.

Depends on. Nothing beyond the cited manuscript.