The shifted convolution problem in function fields

2025/02/22 by Florea, Alexandra, Lalín, Matilde, Malik, Amita +1
#11N37 #11R58 #11R59 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.16067

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 \mathbbF_q[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 r_χ= 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 \mathbbF_q[T]. A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in \mathbbF_q[T] which was not previously available.

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