A geometric perspective on Landau's theorem in function fields

We improve on results obtained by Lior Bary-Soroker, Yotam Smilansky, Adva Wolf in On the Function Field Analogue of Landau's Theorem on Sums of Squares which deals with a function field version of Landau's theorem on the asymptotic number of positive integers $\leq X$ which can be written as a sum of squares. The above paper obtains two asymptotics; one in the large degree limit $n\rightarrow\infty$ and one in the large characteristic $q\rightarrow \infty$. We obtain an expansion of $B_q(n)$ which counts the number of monic polynomials of degree $n$ in $\mathbb{F}_q[T]$ that can be written as $f(T)=|A^2-TB^2|$ that works in the $q^n\rightarrow \infty$ regime. Our approach is done via a twisted Grothendieck Lefschetz trace formula.