A function field analog of Jacobi’s theorem on sums of squares and its moments

Abstract In this article, we establish a function field analog of Jacobi’s theorem on sums of squares and analyze its moments. Our approach involves employing two distinct techniques to derive the main results concerning asymptotic formulas for the moments. The first technique utilizes Dirichlet series framework to derive asymptotic formulas in the limit of large finite fields, specifically when the characteristic of $\mathbb {F}_q[T]$ becomes large. The second technique involves effectively partitioning the set of polynomials of a fixed degree, providing asymptotic formulas in the limit of large polynomial degree.