Research Problems in Function Theory — Problem 2.70

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 $M(r,f)=\max_{|z|=r}|f(z)|$, $T(r,f)=(2\pi)^{-1}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta$, and $\rho(f)=\limsup_{r\to\infty}\log\log M(r,f)/\log r$, where $\log^+t=\max(0,\log t)$. If $E$ is a nonzero entire function with nonzero zeros $a_j$, repeated according to multiplicity, define $\lambda(E)=\inf\{s>0:\sum_j|a_j|^{-s}<\infty\}$, with value $0$ for a finite zero set and $\infty$ when no such $s$ exists. The source update reports counterexamples of Bergweiler and Eremenko for every order greater than $1/2$; this is the historical question, rather than an assertion that the proposed conclusion holds.

Statement recorded in the supplied source.

Let $H$ be transcendental entire with $0<\rho(H)<\infty$ and $\rho(H)\notin\mathbb Z$, and let $f_1,f_2$ be linearly independent entire solutions of \[w''(z)+H(z)w(z)=0.\] Must $E=f_1f_2$ satisfy $\lambda(E)=\infty$? [S2]

Short English statement

Must the product of two independent solutions have zeros of infinite exponent of convergence when the coefficient has finite nonintegral order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302070 (AMR-022-2070). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 2.70 and Update 2.70. https://arxiv.org/abs/1809.07200.