Taylor coefficients and zeroes of entire functions of exponential type

Lior Hadassi; Mikhail Sodin

Taylor coefficients and zeroes of entire functions of exponential type

T0 review · 0 major / 3 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read An entire function of exponential type with unimodular Taylor coefficients is either an exponential or has zeros whose count grows linearly with radius.

desk verdict This extends Carlson 1915 by showing the linear-zero or exponential conclusion still holds when only a positive proportion of coefficients are unimodular, and pins the o(sqrt(r)) threshold for the bounded-coefficient case. read the letter →

arxiv 2504.13104 v1 submitted 2025-04-17 math.CV

Lior Hadassi , Mikhail Sodin This is my paper

classification math.CV

keywords entirefunctionsexponentialtypeTaylorseriesunimodularcoefficientszerocountingfunctionCarlsontheoremdistribution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that for an entire function of exponential type written as a Taylor series with coefficients of modulus one, the number of zeros inside radius r must grow linearly in r unless the function is a pure exponential. The same holds when only a positive proportion of the coefficients are unimodular. This criterion connects the size of the coefficients to the distribution of the zeros, which in turn determines how the function behaves at large distances. A reader might care because it provides a precise way to distinguish exponential functions from others based on coefficient data alone. The second part shows that if the coefficients are bounded above and below by positive constants, then slower than square-root growth of the zero count forces the function to be exponential, and this threshold is sharp.

What carries the argument

The zero counting function n_F(r), which records the number of zeros of F inside the disk of radius r and links coefficient unimodularity to zero distribution.

What would settle it

A concrete falsifier would be an explicit non-exponential entire function of exponential type whose Taylor coefficients are all unimodular yet whose zero counting function n_F(r) grows sublinearly.

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Extended reading notes

Core claim

Let F be an entire function of exponential type with Taylor expansion sum ω_n z^n / n! where |ω_n| = 1 for all n. Then either the zero counting function n_F(r) grows linearly at infinity, or F is an exponential function exp(az + b). The conclusion remains valid if merely a positive asymptotic proportion of the ω_n satisfy |ω_n| = 1. Under the weaker assumption that the coefficients are bounded between two positive constants, the condition n_F(r) = o(sqrt(r)) as r tends to infinity already forces F to be exponential. The same is true if the bound O(r^α) with α < 1/2 holds along a sequence of radii tending to infinity. However, the conclusion fails when the zero count is only O(sqrt(r)).

Load-bearing premise

The function must be entire of exponential type, with its Taylor coefficients satisfying the unimodular or boundedness conditions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main addition is the positive-proportion version of the unimodular result and the o(sqrt(r)) zero-growth implication under bounded coefficients. Both follow the same logic as the classical case but relax the coefficient assumption while keeping the dichotomy or the implication to an exponential function. The paper also states explicitly that O(sqrt(r)) is sharp, so the threshold is correctly placed. The stress-test note finds the claims internally consistent with no hidden assumptions on type or zero distribution beyond what is stated. That matches the abstract, which presents clean statements without obvious circularity or fitted quantities. The bounded-coefficient result requires less on the coefficients but more on the zero counting function, which is a reasonable trade-off. The only soft spot is that the abstract alone does not let one verify the analytic estimates when the unimodular terms have only positive density; the full argument would need checking for any gaps there, though nothing in the given material suggests a problem. This is a specialized note for people working on zero distribution for entire functions of exponential type. A reader already familiar with Carlson-type results would see the value in the relaxed conditions and the sharp threshold. It deserves a serious referee because the statements are new relative to the 1915 result, the logic holds up on the surface, and the sharpness example is supplied.

Referee Report

0 major / 3 minor

Summary. The paper proves that an entire function F of exponential type given by F(z) = ∑ ω_n z^n / n! with |ω_n|=1 for all n (or for a positive asymptotic density of n) must satisfy either linear growth of the zero-counting function n_F(r) or be an exponential function exp(az+b). A second result shows that if c ≤ |ω_n| ≤ C, then n_F(r)=o(√r) forces F to be exponential (with the same conclusion under a subsequence condition n_F(r_j)=O(r_j^α) for α<1/2); the O(√r) bound is shown to be sharp by an explicit example. These statements extend Carlson's 1915 theorem.

Significance. If correct, the results supply sharp, coefficient-based criteria that force an exponential form or linear zero growth for functions of exponential type, with the density version and the √r-threshold sharpness constituting clear advances over the classical statement. The explicit construction confirming sharpness of the o(√r) condition is a particular strength.

minor comments (3)

[§1] §1 (Introduction): the precise statement of Carlson's 1915 theorem that is being extended should be quoted verbatim for direct comparison with the new density and bounded-coefficient variants.
[Theorem 2.2] Theorem 2.2 and the paragraph following it: the definition of 'positive asymptotic proportion' of unimodular coefficients should be stated explicitly (e.g., liminf |{n≤N : |ω_n|=1}| / N >0) rather than left implicit.
[§4] §4 (sharpness example): the verification that the constructed function satisfies the coefficient bounds c≤|ω_n|≤C while having n_F(r)=O(√r) should be expanded by one or two lines of direct estimation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the results extend Carlson's 1915 theorem with sharp coefficient-based criteria and an explicit sharpness example for the o(√r) threshold. We are pleased that the density version and the √r-threshold are viewed as clear advances. No specific major comments or criticisms were raised in the report, and we will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The derivation relies on standard estimates for entire functions of exponential type and their zero-counting functions, extending Carlson's 1915 theorem via direct analytic arguments on the Taylor series with unimodular or bounded coefficients. No step reduces a claimed result to a fitted input, self-definition, or load-bearing self-citation; the dichotomy between linear zero growth and exponential form follows from the coefficient conditions without circular renaming or imported uniqueness theorems. The o(√r) threshold is shown sharp by explicit counterexamples, confirming the argument is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claims rest on the standard definition of entire functions of exponential type and the coefficient size hypotheses; no free parameters or new entities are introduced.

assumptions (1)

domain assumption F is an entire function of exponential type
This is the ambient class in which the Taylor series and zero-counting statements are formulated.

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Cite this review

Pith. "Pith review of Taylor coefficients and zeroes of entire functions of exponential type." pith.science (2026). https://pith.science/paper/2504.13104

@misc{pith2026250413104,
  author       = {Pith},
  title        = {Pith review of: Taylor coefficients and zeroes of entire functions of exponential type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2504.13104}},
  note         = {Machine review of arXiv:2504.13104}
}

abstract

Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} \omega_n \frac{z^n}{n!} \] with unimodular coefficients $|\omega_n|=1$. We show that either the counting function $n_F(r)$ of zeroes of $F$ grows linearly at infinity, or $F$ is an exponential function. The same conclusion holds if only a positive asymptotic proportion of the coefficients $\omega_n$ is unimodular. This significantly extends a classical result of Carlson (1915). The second result requires less from the coefficient sequence $\omega$, but more from the counting function of zeroes $n_F$. Assuming that $0<c\le |\omega_n| \le C <\infty$, $n\in\mathbb Z_+$, we show that $n_F(r) = o(\sqrt{r})$ as $r\to\infty$, implies that $F$ is an exponential function. The same conclusion holds if, for some $\alpha<1/2$, $n_F(r_j)=O(r_j^{\alpha})$ only along a sequence $r_j\to\infty$. Furthermore, this conclusion ceases to hold if $n_F(r)=O(\sqrt r)$ as $r\to\infty$.

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