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

Abstract

We prove an exact formula for the variance of the divisor function over short intervals in F p [T ], where p is a prime integer. We use exponential sums to translate the problem to one involving the ranks of Hankel matrices over finite fields. We prove several results regarding the rank and kernel structure of these matrices, and thus demonstrate their number-theoretic properties. We briefly discuss extending our method to moments higher than the second (the variance); to the k-th divisor function; to correlations of the divisor function with applications to moments of Dirichlet L-functions in function fields; and to the analogous results over F q [T ], where q is a prime power.