Arithmetic constants for symplectic variances of the divisor function

Abstract

Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function $$ over number fields, which were inspired by analogous results over function fields proven by the authors. 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.