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Admissibility of Hörmander--Bernhardsson extremal zeros

Khai-Hoan Nguyen-Dang

TL;DR

We treat the zeros $\{\pm\tau_n\}$ of the Hörmander--Bernhardsson extremal function through the Quine--Heydari--Song framework. By deriving an analytic normal form $\lambda_n=\tau_n^2=(n+\tfrac12)^2+q((n+\tfrac12)^{-2})$, we obtain a full polyhomogeneous heat trace expansion for $\theta(t)=\sum e^{-\tau_n^2 t}$, establishing that $\lambda_n$ is QHS-admissible and enabling meromorphic continuation of the spectral zeta $Z(s)$ and the canonical product $W(\lambda)$. A parity dichotomy shows that even powers $\tau_n^{2p}$ preserve admissibility, while odd powers introduce a nonzero $t\log t$ term, signaling a log-polyhomogeneous obstruction; this leads to a robust perturbation criterion and explains observed residues and special values in Dirichlet-type series $L_+(s)$. Leveraging the HB inversion symmetry, the paper proves Conjecture 1 of BOCRS and connects the small-$t$ heat coefficients to zeta-regularized quantities, with implications for one-dimensional spectral problems and canonical-product asymptotics. The results provide a comprehensive zeta-analytic framework for HB-type spectra and offer a practical criterion for analytic perturbations of quadratic lattices.

Abstract

Let $\varphi$ be the Hörmander--Bernhardsson extremal function, and let $(\pmτ_n)_{n\ge1}$ be its real zeros. Using the recent analytic description of the zero set ${τ_n}$, we prove that the squared zeros $λ_n=τ_n^{2}$ form an admissible sequence in the sense of Quine--Heydari--Song: the heat trace $Θ(t)=\sum_{n\ge1}e^{-λ_n t}$ has a full $t\to0^{+}$ expansion in pure powers of $t^{1/2}$. The proof is based on an analytic normal form \[ λ_n=\Bigl(n+\tfrac12\Bigr)^2+q!\Bigl(\Bigl(n+\tfrac12\Bigr)^{-2}\Bigr), \] a uniform Taylor expansion in $t$, and a Mellin--Hurwitz zeta analysis of the resulting weighted Gaussian sums. As applications we obtain meromorphic continuation and special-value information for the associated spectral zeta function and zeta-regularized product, sharp large-parameter asymptotics for the canonical product $\prod_{n}(1+z/λ_n)$. In particular, we deduce the conjecture by Bondarenko--Ortega-Cerdà--Radchenko--Seip for the special values of the Dirichlet-type series attached to $\varphi$. We also establish a parity dichotomy: sequences $(τ_n^m)$ are QHS--admissible for even $m$, while for odd $m$ a nonzero $t\log t$ term obstructs admissibility.

Admissibility of Hörmander--Bernhardsson extremal zeros

TL;DR

We treat the zeros of the Hörmander--Bernhardsson extremal function through the Quine--Heydari--Song framework. By deriving an analytic normal form , we obtain a full polyhomogeneous heat trace expansion for , establishing that is QHS-admissible and enabling meromorphic continuation of the spectral zeta and the canonical product . A parity dichotomy shows that even powers preserve admissibility, while odd powers introduce a nonzero term, signaling a log-polyhomogeneous obstruction; this leads to a robust perturbation criterion and explains observed residues and special values in Dirichlet-type series . Leveraging the HB inversion symmetry, the paper proves Conjecture 1 of BOCRS and connects the small- heat coefficients to zeta-regularized quantities, with implications for one-dimensional spectral problems and canonical-product asymptotics. The results provide a comprehensive zeta-analytic framework for HB-type spectra and offer a practical criterion for analytic perturbations of quadratic lattices.

Abstract

Let be the Hörmander--Bernhardsson extremal function, and let be its real zeros. Using the recent analytic description of the zero set , we prove that the squared zeros form an admissible sequence in the sense of Quine--Heydari--Song: the heat trace has a full expansion in pure powers of . The proof is based on an analytic normal form a uniform Taylor expansion in , and a Mellin--Hurwitz zeta analysis of the resulting weighted Gaussian sums. As applications we obtain meromorphic continuation and special-value information for the associated spectral zeta function and zeta-regularized product, sharp large-parameter asymptotics for the canonical product . In particular, we deduce the conjecture by Bondarenko--Ortega-Cerdà--Radchenko--Seip for the special values of the Dirichlet-type series attached to . We also establish a parity dichotomy: sequences are QHS--admissible for even , while for odd a nonzero term obstructs admissibility.
Paper Structure (37 sections, 57 theorems, 284 equations)

This paper contains 37 sections, 57 theorems, 284 equations.

Table of Contents

  1. Introduction
  2. The Hörmander--Bernhardsson extremal function and its zero set
  3. Heat traces, zeta regularization, and admissible sequences
  4. Main results: admissibility, parity, and a functional equation for $L_+$
  5. Method: analytic normal forms and Mellin analysis of weighted Gaussian sums
  6. Relation to other works and scope
  7. On notions of admissibility
  8. Future directions
  9. On admissible sequences in the sense of Quine--Heydari--Song
  10. Setting
  11. On conditions (1) and (3)
  12. Reduction to weighted Gaussian sums
  13. Uniform Taylor expansion in $t$
  14. Analytic expansion in $u$ and reduction to $\Theta_m$
  15. Full asymptotics for $\Theta_m(t)$
  16. ...and 22 more sections

Key Result

Theorem 1.1

The heat trace $\theta(t)=\sum_{n\ge1}e^{-\tau_n^2 t}$ admits a full small--time asymptotic expansion as $t\to0^+$ in pure half--integer powers: Consequently, $(\tau_n^2)_{n\ge1}$ is admissible in the sense of QHS.

Theorems & Definitions (116)

  • Theorem 1.1: Heat trace expansion and admissibility
  • Theorem 1.2: Conjecture 1 of BOCRS
  • Theorem 1.3: Parity dichotomy
  • Theorem 1.4: Analytic perturbations of quadratic lattices
  • Definition 2.1: Admissible sequence
  • Remark 2.2
  • Theorem 2.3: Convergent power series representation of $\tau_n$
  • Remark 2.4
  • Lemma 2.5: Lower bound for $\tau_n$
  • proof
  • ...and 106 more