The variance and correlations of the divisor function in Fq [T], and Hankel matrices

We prove an exact formula for the variance of the divisor function over short intervals in A:= Fq [T] A : = F q [ T ] , where q is a prime power; and for correlations of the form $$d(A) d(A+B)$$ d ( A ) d ( A + B ) , where we average both A and B over certain intervals in A A . We also obtain an exact formula for correlations of the form $$d(KQ+N) d (N)$$ d ( K Q + N ) d ( N ) , where Q is prime and K and N are averaged over certain intervals with \,deg\,N ≤ \,deg\,Q -1 ≤ \,deg\,K deg N ≤ deg Q - 1 ≤ deg K ; and we demonstrate that $$d(KQ+N)$$ d ( K Q + N ) and d ( N ) are uncorrelated. We generalize our results to σ z σ z defined by σ z (A):= ∑ E A |A |ᶻ σ z ( A ) : = ∑ E ∣ A | A | z for all monics A ∈ A A ∈ A . Our approach is to use the orthogonality relations of additive characters on Fq F q to translate the problems to ones involving the ranks of Hankel matrices over Fq F q . We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving dₖ d k .