Admissibility of H\"ormander--Bernhardsson extremal zeros

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.