Research Problems in Function Theory — Problem 5.33

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$.A regular triangular lattice has the form $L=a+b\{m+n e^{i\pi/3}:m,n\in\mathbb Z\}$, with $a,b\in\mathbb C$ and $b\ne0$. Let $f:\mathbb D\to\mathbb C\setminus L$ be a holomorphic universal covering projection, and write $f(z)=\sum_{n\ge0}a_nz^n$.

Statement recorded in the supplied source.

For every such lattice and covering projection, must $a_n\to0$ as $n\to\infty$? [S2]

Short English statement

Historical question: do Taylor coefficients of a universal cover of the complement of a triangular lattice tend to zero? The source records an affirmative answer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305033 (AMR-022-5033). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.33, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.