Research Problems in Function Theory — Problem 2.60

Definitions and mathematical statement

Let $f(z)=\sum_{k=0}^{\infty}a_kz^k$ be a nonconstant entire function with $f(z)\ne0$ for all $z\in\mathbb C$. Such a function is necessarily transcendental. For $n\ge1$ let $S_n(z)=\sum_{k=0}^na_kz^k$ denote its Taylor section at the origin. Must it be true that for every $\varepsilon>0$ there are an integer $n\ge1$ and a complex number $z_0$ with \[ S_n(z_0)=0,\qquad |f(z_0)|<\varepsilon? \] Equivalently, the infimum of $|f(z)|$ over the union of the zero sets of all nonconstant Taylor sections should be zero. Constant sections have no zeros because $a_0=f(0)\ne0$ and can simply be ignored. The nonconstant assumption removes the trivial constant counterexample. The function $f$ has no zeros, so the required points only approach small nonzero values of $f$. They cannot remain in a fixed compact set, since $f$ has a positive minimum modulus there. Uniform convergence of sections on compact sets therefore does not settle the problem. The source notes that if the zero-free assumption is dropped and $f$ has a zero, the corresponding assertion follows from Hurwitz's theorem.

Short English statement

Must some zeros of Taylor sections of a nonconstant zero-free entire function occur where the original function is arbitrarily small?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302060 (AMR-022-2060), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.60. Original source indexed by AMR-022-2060. https://arxiv.org/abs/1809.07200.