Research Problems in Function Theory — Problem 2.24

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an entire function $f$, put $Z(f)=\{z\in\mathbb C:f(z)=0\}$ and $O(f)=\{z\in\mathbb C:f(z)=1\}$.

Statement recorded in the supplied source.

Can there exist an entire function $f$ and two distinct straight lines $\ell,m$ such that \[Z(f)\cup O(f)\subseteq\ell\cup m,\] and the combined set $Z(f)\cup O(f)$ has infinitely many points on each of $\ell$ and $m$? [S2] The two-line condition refers to their union: it does not require all zeros to lie on one line and all ones on the other. Update 2.24 makes this distinction explicit through the example $f(z)=\sin(z^2)$, whose zero-points and one-points all lie on the real or imaginary axis. The original paragraph separately discusses Edrei's obstruction to assigning the two levels to two intersecting lines. [S2]

Short English statement

Can all zero-points and one-points of an entire function be confined to two lines, with infinitely many level-set points on each line?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302024 (AMR-022-2024). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.24 and Update 2.24. https://arxiv.org/abs/1809.07200.