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GCP (Gauss' Circle Problem) asks for a closed form for the number of square-lattice points inside a circle, centered at the origin, of radius $r$.

Let's denote by $N(r)$ the number of these points. Then, $N(r)$ is the number of integer solutions (pairs of integers $x$ and $y$) to the inequality

$$x^2+y^2 \le r^2$$

But, what would happen if, instead of setting the center of the circle at the origin, we moved the circle $1/2$ units in the X-axis? The number of lattice points $N^*(r)$ would be the number of integer solutions to

$$(x+1/2)^2+y^2 \le r^2$$

It is easy to show that $N^*(r)$ would also be the number of solutions to

$$x^2+(y+1/2)^2 \le r^2$$

For last, let's denote by $N^{**}(r)$ the number of lattice points of a circle centered at $(1/2, 1/2)$; that is, the number of integer solutions to

$$(x+1/2)^2+(y+1/2)^2 \le r^2$$

Then, my main question is:

Is it possible to get either $N^*(r)$ or $N^{**}(r)$ if only $N(r)$ and $r$ are given?

I found this post about a generalization of this problem, but I would prefer having an exact answer to this question, rather than an approximate result involving big-O notation. It does not mind if any floor function is needed.

Thank you.

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    $\begingroup$ This seems to be a duplicate of your question math.stackexchange.com/questions/2243695/… on MSE, which has been receiving answers and comments $\endgroup$ – Yemon Choi Apr 24 '17 at 17:45
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    $\begingroup$ @Yemom Due to an error in my question, I asked there if there was any direct relationship between the counting functions without knowing $r$ and here about the case $r$ was given. This is a much more difficult problem, so I decided to post it here $\endgroup$ – user3141592 Apr 24 '17 at 17:48
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    $\begingroup$ I thought about editing the question to add that new fact, but, as I already had an answer, I decided to create this new post for not to revive the old one. $\endgroup$ – user3141592 Apr 24 '17 at 17:49
  • $\begingroup$ Since it's clearly possible to compute any of $N(r)$, $N^*(r)$, or $N^{**}(r)$ from just $r$ itself if you allow floor functions and square-root functions (just sum the number of $y$-values for a given $x$, over all possible values of $x$), maybe you could clarify exactly what sort of formula you're looking for. $\endgroup$ – Greg Martin Feb 21 '18 at 16:44
  • $\begingroup$ @GregMartin Well, it would be great to find a closed form for the relationship between these two functions and $N(r)$, but maybe those floor functions and square root can be a good point to start from. $\endgroup$ – user3141592 Feb 21 '18 at 17:18
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I am skeptical, in view of:

Bleher, Pavel M.; Dyson, Freeman J., The variance of the error function in the shifted circle problem is a wild function of the shift, Commun. Math. Phys. 160, No.3, 493-505 (1994). ZBL0808.11058.

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  • $\begingroup$ Thank you very much! That result is interesting (and somewhat disappointing, since it is less probable for me to find the exact answer of the question). On the other hand, I hope that there is some kind of closed form (or maybe involving the floor function) relationship between the original problen and the shifted one for the specific values $\alpha=(0,1/2)$ or $\alpha=(1/2,1/2)$. $\endgroup$ – user3141592 Apr 24 '17 at 20:31

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