Admissibility of Hörmander--Bernhardsson extremal zeros

Abstract

Let $φ$ 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 $φ$. 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.