Admissibility of Hörmander--Bernhardsson extremal zeros

Abstract

Let $\phi$ be the Hörmander--Bernhardsson extremal function, and let $(\pm {\tau }_n{)}_{n\ge 1}$ 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 $\Theta (t)={\sum }_{n\ge 1}{e}^{-{\lambda }_{n}t}$ has a full $t\to 0^{+}$ 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 ${\prod }_n(1+z/{\lambda }_n)$. In particular, we deduce the conjecture by Bondarenko--Ortega-Cerdà--Radchenko--Seip for the special values of the Dirichlet-type series attached to $\phi$. 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.

arXiv comment44 pages, comments welcome!