Abstract
We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of d(f)d(f + h) where f runs over monic polynomials in Fq[T] of a given degree, and h is a given monic polynomial. We prove an asymptotic formula in the range deg (h) < (2 - ϵ) deg (f) . We also consider mixed correlations and self-correlations of rχ = 1 ⋆ χ , the convolution of 1 with a Dirichlet character mod ℓ , where ℓ is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of Fq[T] . A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in Fq[T] which was not previously available.