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

Abstract

In [arXiv:2107.01437], the authors studied the mean-square of certain sums of the divisor function $d_k(f)$ over the function field ${\mathbb{F}}_q[T]$ in the limit as $q\to \infty$ and related these sums to integrals over the ensemble of 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 of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields.

arXiv commentVersion accepted by Transactions of the AMS (with one small change in the statement of Conjecture 1.9)