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In [arXiv:2107.01437], the authors studied the mean-square of certain sums ofthe divisor function $d_k(f)$ over the function field $\mathbb{F}_q[T]$ in thelimit as $q \to \infty$ and related these sums to integrals over the ensembleof symplectic matrices, along similar lines as previous work of Keating,Rodgers, Roditty-Gershon and Rudnick [arXiv:1504.07804] for unitary matrices.We present an analogous problem yielding an integral over the ensemble oforthogonal matrices and pursue a more detailed study of both the symplectic andorthogonal matrix integrals, relating them to symmetric function theory. Thefunction field results lead to conjectures concerning analogous questions overnumber fields.

Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime