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Claim. Theorem 1.1 of the note A Gaussian Construction for an Erdős Problem on Zeros of Successive Derivatives: fix $1/2<\beta<1$, let $\xi_k$ be independent Gaussian coefficients and let

$$f(z)=\sum_{k\ge0}\xi_k\frac{z^k}{(k!)^{1-\beta}}.$$

Then almost every sample of $f$ is a transcendental entire function such that, for every nonempty open $U\subseteq\mathbb{C}$, the derivative $f^{(n)}$ has a zero in $U$ for all sufficiently large $n$. For every increasing sequence $n_1<n_2<\cdots$ the zeros of the $f^{(n_k)}$ are then dense, which is the affirmative answer to the corrected Statement of Problem 906, which asks for a transcendental function. The proof combines a saddle-point estimate for the covariance kernels of the derivatives with a self-contained Offord-type estimate for the probability that a derivative has no zero in a fixed disk.

Submission note. Posted to the site's forum by Przemek Chojecki on 25 April 2026:

Adriano below shared his solution first, so let me share also what I have when it comes to construction. It's to be seen how similar are these approaches.

This is the note done with GPT-5.5 Pro. I wasn't able to formalize it (I don't think it's possible with what is currently in mathlib). The argument uses the cofinite reformulation discussed in the comments here, plus Sodin’s Edelman–Kostlan and Offord-type deviation estimates for Gaussian analytic functions.

Posted to the site's forum by Przemek Chojecki on 1 June 2026:

I wrote the above myself (no AI or automation here), sorry if that text was unclear. Yes, that's a full solution, but as I've mentioned, Adriano posted his solution first.

Postings. The note has no author line. Przemek Chojecki linked it on the site's discussion thread on 25 April 2026, presenting it as done with GPT-5.5 Pro, and on 1 June 2026 Chojecki confirmed on the thread that it is a full solution and that Almeida's construction was posted first. The note is served at an address with no revision to pin. Two replies on the thread report that an automated check found one minor issue in the note; that is not a review.

Order. The note (section 7) gives $f$ the order $1/(1-\beta)>2$, so the function lies outside Theorem 1 of Boas and Reddy (1973), which as printed concerns entire functions of order at most 2 and finite type and conflicts with the other three constructions, as the problem page records.

Standing. No refereed publication, outside review or acceptance by the site was found; the site labels the problem OPEN. The claim stays claimed. The earlier posting of the same day is on Almeida's page.

Depends on. Nothing beyond the cited note.