Research Problems in Function Theory — Problem 7.34

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.

Fix $n\in\mathbb N$ and positive real $\beta_1,\ldots,\beta_n$. For arbitrary $\zeta_1,\ldots,\zeta_n\in\mathbb C$, use the branches of $(1-\zeta_jz)^{\beta_j}$ analytic near zero with value one at zero, and expand \[\prod_{j=1}^n(1-\zeta_jz)^{\beta_j}=1+\sum_{k\ge1}a_kz^k.\] Restrict to choices for which every $a_k$ is real. Let $N(\beta_1,\ldots,\beta_n)$ be the least positive integer such that every admissible choice of the $\zeta_j$ has some $a_k\le0$ with $1\le k\le N$.

Statement recorded in the supplied source.

Find the sharp value, or a good upper bound, for the finite integer $N(\beta_1,\ldots,\beta_n)$ in terms of the exponents, uniformly over all admissible $\zeta_j$. [S2] The source records finiteness and the bound $N\le\beta_1+\cdots+\beta_n+1$ when all exponents are integers. [S2]

Short English statement

Bound the first nonpositive Taylor coefficient of the indicated product uniformly in its complex factors, using only the positive exponents.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307034 (AMR-022-7034). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.34 and its complete update. https://arxiv.org/abs/1809.07200.