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

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.