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

Vivian Kuperberg; Matilde Lalín

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arXiv · 2212.04969

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

Vivian Kuperberg·Matilde Lalín

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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.

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Vivian Kuperberg, Matilde Lalín. 2022-12-09. Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. https://arxiv.org/abs/2212.04969

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