This question is related to this and this ones. The Gauss Circle problem asks for the number $N(r)$ of integer points within a sphere of radius $r$ centered at the origin. It is well known that $N(r) = \pi r^2+ E(r)$, where $|E(r)| \leq 2 \sqrt{2} \pi r$. On the other hand, more sophisticated techniques show that (Van der Corput) $E(r) = O(r^{2/3})$ and (Huxley) $E(r) = O(r^\theta)$, where $\theta$ is something like 0.63.

My question is how "asymptotic" are the more recent results on the problem. Is it possible to estimate the hidden constants behind these results? More precisely, if I were to find $r$ such that

$$\frac{\pi r^2}{\pi r^2+E(r)} > 0.9,$$

what would the best estimate to use? (for the problem above, $|E(r)| \leq 2 \sqrt{2} \pi r$ yields $r$ should be around $40$ while numerical computations suggest that $r > 18$ or $19$ suffices).

[*Huxley paper, given by an answer to this post] Huxley, M. N.(4-WALC) Area, lattice points, and exponential sums. London Mathematical Society Monographs. New Series, 13. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xii+494 pp. ISBN: 0-19-853466-3 11L07 (11J54 11P21)