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 $${\mathcal {A}}:= {\mathbb {F}}_q [T]$$ 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 $${\mathcal {A}}$$ 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 $${{\,\textrm{deg}\,}}N \le {{\,\textrm{deg}\,}}Q -1 \le {{\,\textrm{deg}\,}}K$$ 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 $$\sigma _z$$ σ z defined by $$\sigma _z (A):= \sum _{E \mid A} |A |^z$$ σ z ( A ) : = ∑ E ∣ A | A | z for all monics $$A \in {\mathcal {A}}$$ A ∈ A . Our approach is to use the orthogonality relations of additive characters on $${\mathbb {F}}_q$$ F q to translate the problems to ones involving the ranks of Hankel matrices over $${\mathbb {F}}_q$$ 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 $$d_k$$ d k .