Is there any work on the Gauss circle problem over function fields? I would be thankful if someone had references to provide...
 A: In function fields, the Gauss Circle Problem, at least in its usual formulation, is much simpler than in number fields.
In a number field $K$, one studies the difference $\sum_{n \le x} r_{K}(n) - C_{K} x$, where: $r_K$ is the coefficient of $n^{-s}$ in $\zeta_{K}$, the Dedekind zeta function for $K$, and $C_{K}$ is the residue of $\zeta_K$ at $s=1$. The 'Circle' corresponds to the geometric interpretation of $r_K$ when $K=\mathbb{Q}(i)$. One may also define $r_K$ as
$$r_K(n) = \# \{ I \text{ ideal in }\mathcal{O}_K : \mathrm{Nm}(I)=(n) \}.$$
Now consider a function field $K$ which is a finite field extension of $\mathbb{F}_q(T)$ , the field of rational functions over the finite field with $q$ elements. Let $\mathcal{O}_K$ be the integral closure of $\mathbb{F}_q[T]$ in $K$. We define for any monic $f \in \mathbb{F}_q[T]$:
$$r_{K/\mathbb{F}_q(T)}(f) = \# \{ I \text{ ideal in }\mathcal{O}_K : \mathrm{Nm}_{K/\mathbb{F}_q(T)}(I)=(f)\}.$$
It turns out that in fact we have a closed form formula for $\sum_{f \in \mathbb{F}_q[T]: \deg(f) =n, \text{ monic}}r_{K/\mathbb{F}_q(T)}(f)$. Indeed, consider the (incomplete) Dedekind zeta function
$$\mathcal{Z}_{\mathcal{O}_K}(u) = \prod_{\mathcal{P} \text{ prime in }\mathcal{O}_K} (1-u^{\deg \mathrm{Nm}_{K/\mathbb{F}_q(T)} \mathcal{P}})^{-1} = \sum_{f \in \mathbb{F}_q[T]: \text{ monic}} r_{K/\mathbb{F}_q(T)}(f) u^{\deg f}.$$
Then  $\mathcal{Z}_{\mathcal{O}_K}(u)$ is a rational function, so the $n$-th coefficient has a "finite formula", as explained in Vesselin's comment. For instance, if $K/\mathbb{F}_q(T)$ is a geometric extension, than
$$\mathcal{Z}_{\mathcal{O}_K}(u) = \frac{P(u)}{1-qu}$$
for some polynomial $P$. Thus, for $n \ge \deg P$, we have
$$\sum_{f \in \mathbb{F}_q[T] : \deg(f)=n, \text{ monic}} r_{K/\mathbb{F}_q(T)}(f) =q^n  P(\frac{1}{q}),
$$
and there is no error term in this formula.
Although the Gauss Circle Problem itself was not studied in the function field case AFAIK, norms (and sums of two squares) were indeed studied. There are two common analogues for $\mathbb{Q}(i)$ in the function field case: $\mathbb{F}_q(\sqrt{-T})$ (if $q$ is odd) or $\mathbb{F}_{q^2}(T)$ (works for any $q$). See for instance this work and this one.
A: The Gauss circle problem for lattice points over the algebraic number field $K={\mathbb Q}(\sqrt{-1})$ was considered in Visible lattice points and the Extended Lindelöf Hypothesis (2017).
