The spherical Hall algebra of $\overline{\operatorname{Spec}(\mathcal{O}_K)}$
We generalize a result of M. Kapranov, O. Schiffmann, and E. Vasserot by showing that, for a number field $K$ with class number one, the spherical Hall algebra of $\overline{\operatorname{Spec}(\mathcal{O}_K)}$, where $\mathcal{O}_K$ is the ring of integers of $K$, is isomorphic to the Paley-Wiener shuffle algebra associated to a Hecke $L$-function corresponding to $K$.
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