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

Vivian Kuperberg, Matilde Lalín

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.

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 over the function field in the limit as 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.
Paper Structure (13 sections, 27 theorems, 227 equations, 1 figure)

This paper contains 13 sections, 27 theorems, 227 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some results on determinants
  3. The symmetric function theory context in the symplectic case
  4. The determinant expression in the symplectic case
  5. Some low degree cases
  6. A complex analysis approach in the symplectic case
  7. The Rudnick-Waxman Orthogonal setting
  8. A problem leading to orthogonal distributions
  9. The symmetric function theory context in the orthogonal case
  10. The determinant expression in the orthogonal case
  11. Some low degree cases
  12. A complex analysis approach in the orthogonal case
  13. The conjectures

Key Result

Theorem 1.1

Let $c=\frac{a}{b}$ be a fixed rational number and $k$ be a fixed integer. If $2N$ is a multiple of $b$, then $I_{d_k,2}^S(c2N;N)$ is a polynomial of degree ${2k^2+k-2}$ in $N$.

Figures (1)

  • Figure 1: Semi-standard Young tableau of shape $(6,4,2,2)$

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Remark 1.10
  • ...and 40 more