Sums of divisor functions in $F_{q}[t]$ and matrix integrals

TL;DR

This paper investigates the behavior of divisor functions over function fields, linking mean square sums to matrix integrals and lattice point counts, and proposes conjectures for classical integer problems.

Contribution

It establishes a novel connection between divisor sums in function fields and matrix integrals, providing explicit formulas and conjectures for classical problems.

Findings

Mean square of divisor sums expressed via lattice point counts

Matrix integral over the unitary group relates to divisor functions

Results support conjectures for classical integer problems

Abstract

We study the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements. In the limit as $q\rightarrow\infty$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $d_k(n)$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of certain polynomials, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.