Research Problems in Function Theory — Problem 2.30

Definitions and mathematical statement

For each positive integer $k$, let $S_k\subset\mathbb C$ have no finite limit point; equivalently, its intersection with every compact subset of $\mathbb C$ is finite. For an entire function $f$ and an integer $n\geq0$, $f^{(n)}$ denotes its $n$th complex derivative, with $f^{(0)}=f$.

Question (P. Erdős). For every sequence $(S_k)_{k\geq1}$ of such sets, do there exist a sequence of derivative orders $(n_k)_{k\geq1}$ and an entire function $f$ such that \[ \text{for every }k\geq1\text{ and every }z\in S_k, \qquad f^{(n_k)}(z)=0? \] Only inclusion of $S_k$ in the zero set is required; no assertion that these are all the zeros or that they are simple is made.

The source does not restrict $f$ to be nonzero, nonconstant, or transcendental, and does not require distinct or increasing derivative orders. Those omissions permit vacuous polynomial or identically zero cases and must be resolved before treating the request as a nontrivial open interpolation problem. The conventional derivative orders above are nonnegative integers, but the source does not say whether $0$ is permitted. No missing nontriviality or growth condition is invented. The supplied summary reports the problem open as of the 2018 edition, with no recent resolution located.

Short English statement

Prescribe discrete sets on which selected derivatives of one entire function vanish. The literal statement lacks the nontriviality and derivative-order restrictions needed to identify the intended open problem.

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 2.30, attributed to P. Erdős, https://arxiv.org/abs/1809.07200; source-TeX extraction, accessed 2026-07-29. The 2018 update was retained for status triage.
• Local registry data/unsolvedmath/problems.json, permanent ID $2302030$, source code catalogue entry, Research Problems in Function Theory --- Problem 2.30. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications. No separate result citation is supplied.