Sums of divisor functions and von Mangoldt convolutions in $\mathbb F_q[T]$ leading to symplectic distributions

TL;DR

This paper explores the distribution of divisor functions and von Mangoldt convolutions over function fields, revealing connections to symplectic matrix ensembles and extending prior work on unitary matrices.

Contribution

It introduces new results linking divisor sums and von Mangoldt convolutions in $\,\mathbb{F}_q[T]$ to symplectic matrix distributions, expanding the scope of previous research.

Findings

Established relationships between divisor sums and symplectic distributions.

Extended analysis of von Mangoldt convolutions to symplectic ensembles.

Connected function field problems to random matrix theory.

Abstract

In [arXiv:1504.07804], Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb F_q[T]$ to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.