Arithmetic constants for symplectic variances of the divisor function

TL;DR

This paper explores the arithmetic factors in symplectic variances of the divisor function over number fields, inspired by function field results and random matrix integrals, providing heuristic justifications for these formulas.

Contribution

It offers heuristic explanations for the arithmetic factors in the symplectic variances of divisor sums over number fields, connecting random matrix theory with number theory.

Findings

Heuristic formulas for arithmetic factors in symplectic variances

Connections established between number field results and random matrix integrals

Insights into the role of arithmetic factors in divisor function variances

Abstract

In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function $d_k(n)$ over number fields, which were inspired by analogous results over function fields proven in [arXiv:2107.01437]. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.