I wouldn't expect any of these bounds to give really superb estimates of the last time your ratio achieves a certain record because $E(r)$ and your ratio undergo rapid fluctuations. Also, the exciting places are unlikely to happen at integers. The local minima of $E(r)$ occur at the places where $r=\sqrt{a^2+b^2}.$ Then $E(r)$ increases (at a rate of $2\pi r$) until the next local minimum.

I had a program calculate out to $r=1000.$ There $216342$ values $r=\sqrt{a^2+b^2}$ in that range of which $87483$ have $N(r) \lt \pi r^2.$ and the other $128859$ have $N(r) \gt \pi r^2.$ (I'm not sure why the imbalance. New question!)

This turns out to yield ten guaranteed records with $\frac{\pi r^2 }{N(r)} \gt 1.003.$ They are:

[49, 1.0331], [144, 1.0258], [288, 1.0177], [576, 1.0092], [722, 1.0068], [1152, 1.0061], [1444, 1.0052], [1844, 1.0042], [2592, 1.0037], [2593, 1.0031]

The third entry [288, 1.0177] says that at $r=\sqrt{288}$ the ratio is $1.0177$ and it is never that large again. Most have $r$ an integer or an integer times $\sqrt{2}.$

There are 32 guaranteed records with $\frac{\pi r^2 }{N(r)} \lt 0.997.$ Namely:

[0, 0.], [1, .62832], [2, .69813], [5, .74800], [10, .84908], [13, .90757], [20, .91061], [26, .91777], [29, .93924], [41, .94018], [53, .94070], [65, .95870], [85, .96403], [90, .96499], [130, .97009], [149, .97318], [170, .97995], [185, .98009], [205, .98025], [234, .98149], [340, .98446], [377, .98616], [425, .98683], [586, .98924], [986, .99124], [1325, .99370], [1700, .99473], [1781, .99541], [1885, .99544], [2260, .99593], [3146, .99662], [3400, .99668]

Notice that the last record mentioned is at $r=\sqrt{3400}$ while the calculation went out to $r=\sqrt{1000000}.$ I'm sure that there are many more records in that data.

What I mean by guaranteed is that With the $|\pi r^2 -N(r)|=|E(r)| \lt \sqrt{8}\pi r$ result we can be sure that for all $r \gt 1000$ we have $$0.997179550 \lt \frac{\pi r^2 }{N(r)} \lt 1.0028364498.$$

I wouldn't expect any of these bounds to give really superb estimates of the last time your ratio achieves a certain record because $E(r)$ and your ratio undergo rapid fluctuations. Also, the exciting places are unlikely to happen at integers. The local minima of $E(r)$ occur at the places where $r=\sqrt{a^2+b^2}.$ Then $E(r)$ increases (at a rate of $2\pi r$) until the next local minimum.

I had a program calculate out to $r=1000.$ There $216342$ values $r=\sqrt{a^2+b^2}$ in that range of which $87483$ have $N(r) \lt \pi r^2$ and the other $128859$ have $N(r) \gt \pi r^2.$ (I'm not sure why the imbalance. New question!)

This turns out to yield ten guaranteed records with $\frac{\pi r^2 }{N(r)} \gt 1.003.$ They are:

[49, 1.0331], [144, 1.0258], [288, 1.0177], [576, 1.0092], [722, 1.0068], [1152, 1.0061], [1444, 1.0052], [1844, 1.0042], [2592, 1.0037], [2593, 1.0031]

The third entry [288, 1.0177] says that at $r=\sqrt{288}$ the ratio is $1.0177$ and it is never that large again. Most have $r$ an integer or an integer times $\sqrt{2}.$

There are 32 guaranteed records with $\frac{\pi r^2 }{N(r)} \lt 0.997.$ Namely:

[0, 0.], [1, .62832], [2, .69813], [5, .74800], [10, .84908], [13, .90757], [20, .91061], [26, .91777], [29, .93924], [41, .94018], [53, .94070], [65, .95870], [85, .96403], [90, .96499], [130, .97009], [149, .97318], [170, .97995], [185, .98009], [205, .98025], [234, .98149], [340, .98446], [377, .98616], [425, .98683], [586, .98924], [986, .99124], [1325, .99370], [1700, .99473], [1781, .99541], [1885, .99544], [2260, .99593], [3146, .99662], [3400, .99668]

Notice that the last record mentioned is at $r=\sqrt{3400}$ while the calculation went out to $r=\sqrt{1000000}.$ I'm sure that there are many more records in that data.

What I mean by guaranteed is that With the $|\pi r^2 -N(r)|=|E(r)| \lt \sqrt{8}\pi r$ result we can be sure that for all $r \gt 1000$ we have $$0.997179550 \lt \frac{\pi r^2 }{N(r)} \lt 1.0028364498.$$

I wouldn't expect any of these bounds to give really superb estimates of the last time your ratio achieves a certain record because $E(r)$ and your ratio undergo rapid fluctuations. Also, the exciting places are unlikely to happen at integers.

The last integer for which $\left| \frac{\pi r^2 }{\pi r^2 + E(r)} - 1 \right| \gt 10^{-2}$ is $r=12.$ However $N(\sqrt{586})=1861$ while $586\pi\approx 1840.97$ giving a ratio of $0.98923=1-0.01076.$ If you change $0.01$ to $0.0132$ the record is at $r=\sqrt{425}.$ But if you change it to $0.0082$ it is at $r=\sqrt{986}.$

Are you considering all positive real $r$ or all$ \sqrt{a^2+b^2}$ or only integers? The local minima of $E(r)$ occur at the places where $r=\sqrt{a^2+b^2}.$ Then $E(r)$ increases (at a rate of $2\pi r$) until the next local minimum.

For $r=\sqrt{281},\sqrt{288},\sqrt{289}$ one has $N(r)=885,889,901$ and the last time the ratio is as large as $1.01$ is at $\sqrt{288}$ (or $\sqrt{288}-\varepsilon$ if allowed) with ratio of $1.0177$ (or $1.0223$ .)

I wouldn't expect any of these bounds to give really superb estimates of the last time your ratio achieves a certain record because $E(r)$ and your ratio undergo rapid fluctuations. Also, the exciting places are unlikely to happen at integers. The local minima of $E(r)$ occur at the places where $r=\sqrt{a^2+b^2}.$ Then $E(r)$ increases (at a rate of $2\pi r$) until the next local minimum.

I had a program calculate out to $r=1000.$ There $216342$ values $r=\sqrt{a^2+b^2}$ in that range of which $87483$ have $N(r) \lt \pi r^2.$ and the other $128859$ have $N(r) \gt \pi r^2.$ (I'm not sure why the imbalance. New question!)

This turns out to yield ten guaranteed records with $\frac{\pi r^2 }{N(r)} \gt 1.003.$ They are:

[49, 1.0331], [144, 1.0258], [288, 1.0177], [576, 1.0092], [722, 1.0068], [1152, 1.0061], [1444, 1.0052], [1844, 1.0042], [2592, 1.0037], [2593, 1.0031]

The third entry [288, 1.0177] says that at $r=\sqrt{288}$ the ratio is $1.0177$ and it is never that large again. Most have $r$ an integer or an integer times $\sqrt{2}.$

There are 32 guaranteed records with $\frac{\pi r^2 }{N(r)} \lt 0.997.$ Namely:

[0, 0.], [1, .62832], [2, .69813], [5, .74800], [10, .84908], [13, .90757], [20, .91061], [26, .91777], [29, .93924], [41, .94018], [53, .94070], [65, .95870], [85, .96403], [90, .96499], [130, .97009], [149, .97318], [170, .97995], [185, .98009], [205, .98025], [234, .98149], [340, .98446], [377, .98616], [425, .98683], [586, .98924], [986, .99124], [1325, .99370], [1700, .99473], [1781, .99541], [1885, .99544], [2260, .99593], [3146, .99662], [3400, .99668]

Notice that the last record mentioned is at $r=\sqrt{3400}$ while the calculation went out to $r=\sqrt{1000000}.$ I'm sure that there are many more records in that data.

What I mean by guaranteed is that With the $|\pi r^2 -N(r)|=|E(r)| \lt \sqrt{8}\pi r$ result we can be sure that for all $r \gt 1000$ we have $$0.997179550 \lt \frac{\pi r^2 }{N(r)} \lt 1.0028364498.$$