The Variance and Correlations of the Divisor Function in $\mathbb{F}_q [T]$, and Hankel Matrices

TL;DR

This paper derives exact formulas for the variance and correlations of the divisor function over function fields, linking these to Hankel matrix ranks and kernels, with implications for moments of Dirichlet L-functions.

Contribution

It introduces a novel method connecting divisor function correlations to Hankel matrix properties over finite fields, providing explicit formulas and uncorrelation results.

Findings

Exact formulas for divisor function variance in function fields.

Correlation formulas involving divisor functions and Hankel matrices.

Uncorrelation of certain divisor function pairs in specified ranges.

Abstract

We prove an exact formula for the variance of the divisor function over short intervals in $\mathcal{A} := \mathbb{F}_q [T]$, where $q$ is a prime power. A slight adaption of the proof allows us to obtain an exact formula for correlations of the form $d(A) d(A+B)$, where we average both $A$ and $B$ over certain intervals in $\mathcal{A}$. We also consider correlations of the form $d(KQ+N) d (N)$, where $Q$ is prime and $K$ and $N$ are averaged over certain intervals. If $\mathrm{deg } K < \mathrm{deg } Q -1$, then these correlations appear in the off-diagonal terms for the fourth moment of Dirichlet $L$-functions. We consider the case $\mathrm{deg } K \geq \mathrm{deg }Q -1$ and obtain an exact formula for the correlations. Further, we demonstrate that $d(KQ+N)$ and $d (N)$ are uncorrelated for the given ranges of $K$ and $N$. Our approach to these problems is to use the orthogonality relations of additive characters on $\mathbb{F}_q$ to translate the problems to ones involving the ranks of Hankel matrices over $\mathbb{F}_q$. Most of the paper is dedicated to proving several results regarding the rank and kernel structure of these matrices, and thus demonstrating their number-theoretic properties. We briefly discuss extending our method to moments higher than the second (the variance) over intervals; to the $k$-th divisor function; and to correlations of the divisor function with applications to moments of Dirichlet $L$-functions in function fields.