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Consider a generalization of the Gauss circle problem and let $$ N(r):= \#\lbrace (x,y,z)\in \Bbb Z^3_{\gt 0}\vert \ln^2(x)+\ln^2(y)+\ln^2(z)\le\ln(r) \rbrace $$

I found that

$$N(r)=\left(\frac{2\pi}{\sqrt3}\right)^{3/2} I_{3/2}\!\left(\sqrt{3\ln r}\right)+E(r)$$

where the error term, $E(r)=O( (\ln r)^k)$ for some $k>0$, and $I$ is a Bessel function.

Can the error term $E(r)$ be improved?

The Bessel term comes from the enclosed volume of the log sphere and the error term $E(r)$ grows like $O((\ln r)^k)$, but I think it can be improved significantly.

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    $\begingroup$ Why do you call this "counting lattice points inside an ellipsoid"? You're basically talking about lattice points inside a sphere (up to your use of $\ln$ and that you want all coordinates positive). $\endgroup$ Commented Jun 16 at 18:31
  • $\begingroup$ @SamHopkins It was for lack of a better term. I can change it $\endgroup$ Commented Jun 16 at 18:33
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    $\begingroup$ And for that matter why are you taking $\ln$ of the coordinates at all? $\endgroup$ Commented Jun 16 at 18:34
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    $\begingroup$ Yeah this doesn't deserve a downvote but whatever. $\endgroup$ Commented Jun 16 at 20:07
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    $\begingroup$ @JohnMcManus : (i) How did you get $E(r)=O( (\ln r)^k)$? (ii) How did you get "the enclosed volume of the log sphere"? $\endgroup$ Commented Jun 17 at 21:08

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It seems highly unlikely that the bound $E(r)=O(\ln^k r)$ can be improved. Moreover, it seems highly unlikely that the bound $E(r)=O(\ln^k r)$ is true.

Indeed, let $$B_r:=\{ (x,y,z)\in\Bbb Z^3_{>0}\colon \ln^2x+\ln^2y+\ln^2z\le R^2 \},$$ where $R:=\sqrt{\ln r}$. Then, for each natural number $x\in[1,e^R\,]$ and $y_x:=\lfloor\exp\sqrt{R^2-\ln^2x}\rfloor$, we have $(x,y_x,1)\in B_r$ but $(x,y_x+1,1)\notin B_r$, so that the point $(x,y_x,1)$ is in the "lattice boundary" of the set $B_r$, and there are $\sim e^R=e^{\sqrt{\ln r}}$ such "lattice boundary" points. So, one cannot expect $E(r)$ to be much smaller than $e^{\sqrt{\ln r}}$, which latter is much greater than $\ln^k r$ for any real $k$ and all large $r$.

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