Problem 229

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claims/: The 1 claim page of Problem 229, one per claimant's result; the problem's standing derives from them.

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Statement. Let $(S_n)_{n\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n\geq 1$, there exists some $k_n\geq 0$ such that

$$f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?$$

Status. PROVED (LEAN). The site credits Barth and Schneider [BaSc72] with the affirmative answer; the claim page [[problems/analysis/E0229/claims/1972_03_01_barth_schneider|Barth and Schneider 1972]] records it, accepted on its refereed publication and the site's credit. The site's Lean qualification refers to Boris Alexeev's formalization of that solution, linked from the claim page and not built here.

Source. erdosproblems.com/229, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #229, https://www.erdosproblems.com/229.

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.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.

Formalization. Statement in formal-conjectures at the revision current on 2026-10-06, whose formal_proof attribute points at a Lean 4 proof in Boris Alexeev's lean-proofs repository, recorded on the claim page as a formalization of Barth and Schneider's solution; this project has not built it.

Current assessment

The site's formulation of 2026-09-04 asks, for a sequence of sets $S_n$ of complex numbers without finite limit points, for one transcendental entire function with, for each $n$, some derivative vanishing on all of $S_n$. Barth and Schneider's 1972 paper constructs such a function, with every derivative order positive, so the question is answered yes. The standing rests on the refereed paper and the site's credit, as the claim page records. The paper is not held in the library, so no proof is compiled or reviewed here, and the Lean development the site's label refers to has not been built or audited by this project. No status search beyond the site is recorded.