Consider the set of coprime integer pairs $\mathcal{C} \subset \mathbb{Z}^2$ and the circle of radius $r$ centered at the origin. The function $$f(r) = \min_{(m,n) \in \mathcal{C}} \bigl| \sqrt{m^2 + n^2} - r \bigr|$$ represents the minimum Euclidean distance between $\mathcal{C}$ and the circle of radius $r$.

I am interested in an explicit upper bound on $f(r)$. An obvious upper bound is $f(r) \leq \frac{1}{2}$ for $r \geq 1$, since circles of radius at least $1$ will cross the line $(1,\mathbb{Z})$ which consists of consecutive coprime pairs. But I need way better than this, ideally a bound that tends to zero as $r \to \infty$.

First of all, is it even true that $\limsup_{r \to \infty} f(r) = 0$?

The classical result on the asymptotic density of coprimes tending to $6/\pi^2$ would indeed suggest this, but the statement seems way too weak to be of any use here. I am also aware of the Jacobsthal function, though bounds thereof do not seem to help either for the question at hand.

**NB:** in my actual research problem I consider centered ellipses rather than circles, which might make the problem even harder. Specifically, I want to find a good upper bound on
$$\min_{(m,n) \in \mathcal{C}} \left| \begin{bmatrix} m & n \end{bmatrix} \mathbf{A} \begin{bmatrix} m \\ n \end{bmatrix} - C \right|$$
for some $C > 0$ and positive definite $\mathbf{A} \in \mathbb{R}^{2 \times 2}$. This bound should tend to zero as $C \to \infty$.

insidea circle. However, in my problem I do not need bounds on the $O(r^{1+\epsilon})$ term (the typically studied problem), but lower bounds onincrementsof this error term. $\endgroup$