Research Problems in Function Theory — Problem 5.37

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 $0$.Let $1\le n_0<n_1<\cdots$ be integers and let $f(z)=\sum_{k\ge0}c_kz^{n_k}$ be holomorphic on $\mathbb D$, with $\sum_{k\ge0}|c_k|=\infty$. For real $\alpha<\beta$ with $0<\beta-\alpha\le2\pi$, let $V_{\alpha,\beta}=\{re^{i\theta}:0<r<1,\ \alpha<\theta<\beta\}$ be an open angular sector.Do not assume Hadamard gaps. Put $M(r,f)=\max_{|z|=r}|f(z)|$, \[\mu=\limsup_{r\uparrow1}\frac{\log\log M(r,f)}{-\log(1-r)},\qquad N^0(t)=\#\{k:n_k\le t\}. \] Suppose for some $0<\gamma<1$ that $N^0(t)=O(t^{1-\gamma})$ as $t\to\infty$ and $0<\mu<(1-\gamma)/\gamma$. The positive growth index makes the double logarithm well-defined for $r$ sufficiently near $1$.

Statement recorded in the supplied source.

For every finite value $a\in\mathbb C$, must (i) $f(z)=a$ have at least one solution in $\mathbb D$; (ii) it have infinitely many distinct solutions in $\mathbb D$; and (iii) it have at least one solution in every sector $V_{\alpha,\beta}$? These are three separate assertions.Investigate in particular the sequences $n_k=\lfloor(k+1)^\eta\rfloor$ with $1<\eta<3/2$, subject to the stated counting and growth conditions. [S2]

Short English statement

For sparse gap series in the intermediate positive-growth range, ask the same three value-distribution questions, especially for exponents growing like k to a power between one and three halves.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305037 (AMR-022-5037). 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 5.37, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.