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Admissibility of Hörmander--Bernhardsson extremal zeros

The paper proves that the squared real zeros of the Hörmander–Bernhardsson extremal function form an admissible sequence by utilizing a new analytic normal form and Mellin–Hurwitz zeta analysis, thereby resolving a conjecture regarding their associated Dirichlet-type series and establishing a parity dichotomy for the admissibility of their powers.

Original authors: Khai-Hoan Nguyen-Dang

Published 2026-02-12

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Khai-Hoan Nguyen-Dang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master clockmaker. You have a collection of incredibly precise, high-end watches. You want to know if these watches are "perfect"—meaning, if you were to look at them through a microscope, would their tiny gears follow a smooth, predictable pattern, or would there be a strange, jittery "glitch" that ruins their mathematical elegance?

This paper is essentially a high-level mathematical investigation into whether a specific set of "mathematical gears" (called the Hörmander–Bernhardsson zeros) is smooth or glitchy.

Here is the breakdown of the story:

1. The "Gears": The H-B Zeros

In complex analysis, there are certain "extremal functions"—think of these as the most efficient, perfectly shaped curves possible under certain rules. These curves have "zeros" (points where the curve hits zero). These zeros are like the teeth on a gear.

For a long time, mathematicians knew these teeth were almost perfectly spaced, like the marks on a ruler. But "almost" isn't good enough for high-level math. We need to know if the tiny deviations from the ruler follow a pattern that is "clean" or "messy."

2. The Test: "Admissibility" (The Smoothness Test)

The author uses a concept called Admissibility.

Imagine you are heating up a metal rod. As it heats up, it vibrates. In math, we use something called a Heat Trace to measure these vibrations.

An "Admissible" sequence is like a perfectly tuned bell: when you strike it, the sound decays in a very predictable, smooth way (like a pure musical note).
A "Non-admissible" sequence is like a bell with a tiny crack: when you strike it, you hear a strange, ghostly "hiss" or a "logarithmic glitch" that shouldn't be there.

3. The Discovery: The Parity Dichotomy (The "Even vs. Odd" Rule)

This is the "Aha!" moment of the paper. The author discovered that whether these mathematical gears are "smooth" or "glitchy" depends entirely on whether you square them or not.

The Even Rule (The Smooth Path): If you take these zeros and square them (or raise them to any even power like 2, 4, 6...), the "glitch" disappears. The math becomes "Admissible." It’s like taking a bumpy road and smoothing it out with a layer of perfect asphalt. Everything becomes predictable, and we can use powerful tools to calculate "special values" (the mathematical equivalent of finding the exact center of gravity).
The Odd Rule (The Glitchy Path): If you leave them as they are (power of 1) or raise them to odd powers (3, 5, 7...), the "glitch" remains. A strange mathematical term—a logarithm—appears in the vibrations. This "hiss" prevents the sequence from being "Admissible." It’s like trying to play a violin with a string that has a tiny, irregular knot in it.

4. Why does this matter? (The Big Picture)

By proving this "Even vs. Odd" rule, the author did two major things:

Solved a Mystery: They proved a long-standing conjecture (a mathematical "guess") regarding how these functions behave.
Built a Toolkit: They provided a "recipe" (an analytic normal form) that other mathematicians can use to study different types of mathematical "gears" in physics and geometry.

Summary in a Nutshell

The paper proves that the "teeth" of a very special mathematical shape are perfectly behaved if you look at them in pairs (even powers), but they carry a tiny, beautiful, "glitchy" secret if you look at them individually (odd powers). This distinction allows mathematicians to finally master the "music" produced by these complex functions.

This paper, titled "Admissibility of Hörmander–Bernhardsson Extremal Zeros" by Khai-Hoan Nguyen-Dang, provides a rigorous analytic treatment of the zero set of the Hörmander–Bernhardsson extremal function. The work bridges the gap between classical extremal problems in complex analysis and the spectral theory of zeta-regularized products.

1. The Problem

The central object of study is the Hörmander–Bernhardsson extremal function $\phi$ , an even entire function of exponential type $\pi$ that minimizes the $L^1$ -norm on the real line subject to $\phi(0)=1$ . The zeros of this function, denoted $\{\pm\tau_n\}_{n \geq 1}$ , are real and simple.

While recent works (notably Bondarenko–Ortega-Cerdà–Radchenko–Seip, [BOCRS25]) have established that these zeros are $\ell^2$ -perturbations of the half-integer lattice ( $n + 1/2$ ), this proximity does not automatically guarantee "spectral" behavior. Specifically, the author investigates whether the sequences derived from these zeros satisfy the Quine–Heydari–Song (QHS) admissibility criterion. QHS admissibility is a strict condition requiring the heat trace $\Theta(t) = \sum e^{-\lambda_n t}$ to have a full asymptotic expansion in pure powers of $t$ as $t \to 0^+$ , specifically forbidding logarithmic terms ( $t \log t$ ).

2. Methodology

The author employs a sophisticated combination of analytic normal forms and Mellin–Hurwitz zeta analysis:

Analytic Normal Form: The author proves that while the zeros $\tau_n$ themselves contain a "borderline" $1/n$ correction that would produce a $t \log t$ term in the heat trace, the squared zeros $\lambda_n = \tau_n^2$ can be reparametrized into an analytic normal form: $\lambda_n = (n + a)^2 + q((n + a)^{-2})$ , where $q$ is analytic. This squaring "cancels" the problematic harmonic weight.
Mellin–Hurwitz Analysis: To derive the small-time asymptotics, the author uses the Mellin transform to relate the heat trace to the Hurwitz zeta function. By shifting the contour of integration and analyzing the residues of the Gamma function and the Hurwitz zeta function, the author extracts the coefficients of the asymptotic expansion.
Coefficient Extraction: The author utilizes the relationship between the small-time heat expansion and the large- $\lambda$ expansion of the canonical product $W(\lambda)$ via Watson-type lemmas for Laplace transforms.

3. Key Contributions and Results

A. Admissibility of Squared Zeros (Theorem 1.1)

The primary result is that the sequence of squared zeros $(\tau_n^2)_{n \geq 1}$ is QHS-admissible. The heat trace $\theta(t)$ admits a full asymptotic expansion in pure half-integer powers of $t$ .

B. Proof of the BOCRS Conjecture (Theorem 1.2)

Using the admissibility of $\lambda_n$ , the author proves a major conjecture from [BOCRS25] regarding the symmetry of the Dirichlet-type series $L^+(s) = \sum \tau_n^{-s}$ . Specifically, for every $k \in \mathbb{Z}$ :
$L^+(-2k) = \frac{L^+(2k)}{(2\pi i C)^{2k}}$
This is achieved by identifying the coefficients of the inversion identity for $\phi$ with the special values of the spectral zeta function at non-positive integers.

C. The Parity Dichotomy (Theorem 1.3)

The author establishes a sharp distinction between even and odd powers of the zeros:

Even powers $(\tau_n^{2m})$ : These are always QHS-admissible, leading to "clean" spectral theory (meromorphic continuation of zeta functions and pure-power asymptotics for canonical products).
Odd powers $(\tau_n^{2m+1})$ : These are not admissible. The heat trace contains a nonzero $t \log t$ term. This logarithmic term is explicitly linked to the residues of the Dirichlet series $L^-(s)$ .

D. General Admissibility Criterion (Theorem 1.4)

The paper provides a robust tool for the mathematical community: any sequence that is an analytic perturbation of a quadratic lattice (in the form $\lambda_n = (n+a)^2 + q((n+a)^{-2})$ ) is QHS-admissible.

4. Significance

This paper is significant for several reasons:

Theoretical Unification: It connects the theory of extremal entire functions (Paley–Wiener space) with the calculus of zeta-regularized determinants used in mathematical physics and spectral geometry.
Resolution of Conjectures: It settles a key conjecture regarding the functional equation of Dirichlet series associated with extremal problems.
New Analytic Framework: By identifying the "parity dichotomy," the author provides a roadmap for understanding why certain spectral sequences behave like elliptic operators (pure powers) while others exhibit logarithmic singularities (log-polyhomogeneous expansions).
Broad Applicability: The general criterion for analytic perturbations of quadratic lattices has immediate applications to Sturm–Liouville problems, Bessel-type zeros, and de Branges sampling theory.