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.

Edit: Try to fix the question so that it makes sense.

My question is how "asymptotic" are the more recent results on the problem. Is it possible to estimate the hidden constants behind these results? For example, we know that

$$\lim_{r \to \infty} \frac{\pi r^2}{\pi r^2+E(r)} = 1.$$

If I were to find $r_0$ such that the for $r \geq r_0$ the "relative error" of the limit is $$\left| \frac{\pi r^2 }{\pi r^2 + E(r)} - 1 \right| \leq 10^{-2} $$ what would be the best estimate to use? (for the problem above, $|E(r)| \leq 2 \sqrt{2} \pi r$ gives us that $r$ should be greater than $280$ while numerical computations suggest that $r > 13$ 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)

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.

Edit: Try to fix the question so that it makes sense.

My question is how "asymptotic" are the more recent results on the problem. Is it possible to estimate the hidden constants behind these results? For example, we know that

$$\lim_{r \to \infty} \frac{\pi r^2}{\pi r^2+E(r)} = 1.$$

If I were to find $r_0$ such that the for $r \geq r_0$ the "relative error" of the limit is $$\left| \frac{\pi r^2 }{\pi r^2 + E(r)} - 1 \right| \leq 10^{-2} $$ what would be the best estimate to use? (for the problem above, $|E(r)| \leq 2 \sqrt{2} \pi r$ gives us that $r$ should be greater than $280$ while numerical computations suggest that $r > 13$ 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)

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)

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.

Edit: Try to fix the question so that it makes sense.

My question is how "asymptotic" are the more recent results on the problem. Is it possible to estimate the hidden constants behind these results? For example, we know that

$$\lim_{r \to \infty} \frac{\pi r^2}{\pi r^2+E(r)} = 1.$$

If I were to find $r_0$ such that the for $r \geq r_0$ the "relative error" of the limit is $$\left| \frac{\pi r^2 }{\pi r^2 + E(r)} - 1 \right| \leq 10^{-2} $$ what would be the best estimate to use?  (for the problem above, $|E(r)| \leq 2 \sqrt{2} \pi r$ gives us that $r$ should be greater than $280$ while numerical computations suggest that $r > 13$ 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)