Research Problems in Function Theory — Problem 7.21

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 infinite sequence $(z_k)_{k\ge1}$ on the complex unit circle and $\ell\in\mathbb N$, set \[A_\ell=\limsup_{m\to\infty}\left|\sum_{k=1}^m z_k^\ell\right|\in[0,+\infty].\] The assertion $A_\ell=o(\ell)$ means that $A_\ell/\ell\to0$ as the positive integer $\ell$ tends to infinity.

Statement recorded in the supplied source.

Does there exist a sequence $(z_k)_{k\ge1}$ with $|z_k|=1$ for every $k$ and $A_\ell=o(\ell)$? [S2] The source compares this target with an achievable linear upper bound and Clunie's square-root lower bound for infinitely many exponents. [S2]

Short English statement

Can one choose unit complex numbers so that the limiting sizes of all their power sums grow sublinearly in the exponent?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307021 (AMR-022-7021). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.21 and its complete update. https://arxiv.org/abs/1809.07200.