On the Function Field Analogue of Landau's Theorem on Sums of Squares

TL;DR

This paper explores the function field analogue of Landau's theorem, estimating the density of sums of two squares in polynomial rings over finite fields, and finds consistent asymptotic behaviors in different limit regimes.

Contribution

It introduces a new analogue of sums of two squares in function fields and analyzes their asymptotic densities in two distinct limit cases.

Findings

Asymptotic density estimates for sums of two squares in function fields.

Consistent limiting behavior across different asymptotic regimes.

Extension of classical number theory results to polynomial rings over finite fields.

Abstract

This paper deals with function field analogues of the famous theorem of Landau which gives the asymptotic density of sums of two squares in $\mathbb{Z}$. We define the analogue of a sum of two squares in $\mathbb{F}_q[T]$ and estimate the number $B_q(n)$ of such polynomials of degree $n$ in two cases. The first case is when $q$ is large and $n$ fixed and the second case is when $n$ is large and $q$ is fixed. Although the methods used and main terms computed in each of the two cases differ, the two iterated limits of (a normalization of) $B_q(n)$ turn out to be exactly the same.