Admissibility of H\"ormander--Bernhardsson extremal zeros

TL;DR

This paper proves that the squared zeros of the H"ormander--Bernhardsson extremal function form an admissible sequence with a full small-time heat trace expansion, and explores implications for spectral zeta functions and related conjectures.

Contribution

It establishes the admissibility of zeros for the extremal function and derives detailed spectral and asymptotic properties, confirming a conjecture and revealing a parity-based dichotomy.

Findings

Squared zeros form an admissible sequence with a pure power heat trace expansion.

Derived meromorphic continuation and special values for the spectral zeta function.

Confirmed the conjecture on special values of Dirichlet-type series associated with the extremal function.

Abstract

Let $\varphi$ be the H\"ormander--Bernhardsson extremal function, and let $(\pm\tau_n)_{n\ge1}$ be its real zeros. Using the recent analytic description of the zero set ${\tau_n}$, we prove that the squared zeros $\lambda_n=\tau_n^{2}$ form an admissible sequence in the sense of Quine--Heydari--Song: the heat trace $\Theta(t)=\sum_{n\ge1}e^{-\lambda_n t}$ has a full $t\to0^{+}$ expansion in pure powers of $t^{1/2}$. The proof is based on an analytic normal form \[ \lambda_n=\Bigl(n+\tfrac12\Bigr)^2+q!\Bigl(\Bigl(n+\tfrac12\Bigr)^{-2}\Bigr), \] 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 $\prod_{n}(1+z/\lambda_n)$. In particular, we deduce the conjecture by Bondarenko--Ortega-Cerd\`a--Radchenko--Seip for the special values of the Dirichlet-type series attached to $\varphi$. We also establish a parity dichotomy: sequences $(\tau_n^m)$ are QHS--admissible for even $m$, while for odd $m$ a nonzero $t\log t$ term obstructs admissibility.