---

Claim. Let $\xi_0,\xi_1,\ldots$ be independent standard complex Gaussian random variables and let

$$F(z)=\sum_{k\ge0}\xi_k\frac{z^k}{\sqrt{k!}}$$

be the planar Gaussian entire function. 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.

Argument. The manuscript first reduces the density condition to this cofinite property over a countable family of disks. The Edelman-Kostlan formula for the expected number of zeros of a Gaussian analytic function, together with Plancherel-Rotach asymptotics for Laguerre polynomials, gives an expected zero count of order $\sqrt n$ for $F^{(n)}$ in each fixed disk away from the origin. An Offord-type estimate bounds the probability that $F^{(n)}$ has no zero in such a disk by $C\exp(-c\sqrt n)$, and the Borel-Cantelli lemma over disks with rational centers and radii finishes the proof. The manuscript makes its conclusion subject to the standard estimates it cites.

Postings. The claimant, Adriano Almeida, gave the argument on the site's discussion thread on 25 April 2026 and linked the manuscript, A Probabilistic Construction for the Transcendental Form of Erdős Problem 906, later the same day; the preprint link is pinned to the repository's state at that posting. The manuscript's byline names Almeida as submitter and says that the draft was prepared with AI assistance; the PDF's document metadata names ChatGPT. Two replies on the thread report that an automated check found one minor issue in the manuscript; that is not a review.

A conflicting theorem. The function $F$ has order 2 and type $1/2$ almost surely, so it falls under Theorem 1 of Boas and Reddy (Bull. Amer. Math. Soc. 79 (1973), 64-65), which, as printed, gives every transcendental entire function of order at most 2 and finite type arbitrarily large disks on each of which infinitely many of its derivatives have no zero. If the claim is correct, that theorem is false as printed; the manuscript does not address it. The conflict is recorded on the problem page.

Standing. No refereed publication, outside review or acceptance by the site was found; the site labels the problem OPEN. The claim stays claimed. The other constructions are on Chojecki's page, Hou's page and He's page.

Depends on. Nothing beyond the cited manuscript.