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

TL;DR

This paper explores conjectures connecting sums of divisor functions over function fields to matrix integrals over symplectic and orthogonal groups, providing new insights and conjectures for number fields.

Contribution

It introduces an orthogonal matrix integral approach to divisor sums, extending previous symplectic results and relating them to symmetric function theory.

Findings

Derived orthogonal matrix integrals related to divisor sums

Connected matrix integrals to symmetric function theory

Formulated conjectures for number field analogues

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.