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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 question is: Is there any direct relationship between $N(r)$, $N^*(r)$ and $N^{**}(r)$ ?

Thank you.

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  • $\begingroup$ @Matthew I have no idea about how to attack this problem analytically, so I tried a few elementary geometric things, with no success $\endgroup$
    – user3141592
    Commented Apr 20, 2017 at 22:23
  • $\begingroup$ Have you tried calculating $N(r), N^*(r)$ and $N^{**}(r)$ for a few $r$ values to see if you can make any conjectures? $\endgroup$
    – Matthew Conroy
    Commented Apr 20, 2017 at 23:21
  • $\begingroup$ Also, you might try working through some theorems about $N(r)$, to see if you could modify them to be applicable to $N^*$ or $N^{**}$. $\endgroup$
    – Matthew Conroy
    Commented Apr 20, 2017 at 23:23
  • $\begingroup$ @Matthew For example, which Theorems? $\endgroup$
    – user3141592
    Commented Apr 21, 2017 at 5:31
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    $\begingroup$ You might want to look at: projecteuclid.org/euclid.cmp/1104253074 $\endgroup$
    – Matthew Conroy
    Commented Apr 21, 2017 at 20:30

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(Expanding on my comment.)

The answer to your second question is "no".

I find $N(3.3)=N(3.4)=37$ while $N^*(3.3)=34$ and $N^*(3.4)=38$.

Assuming my calculations are correct, this proves that you cannot know $N^*(r)$ "only with the value of $N(r)$". In particular, if we know that $N(r)=37$, we cannot conclude the value of $N^*(r)$. In other words, $N^*(r)$ is not a function of $N(r)$.

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    $\begingroup$ That is not a function of $N(r)$: you are using the value of $N(r)$ and $r$. This is not what your question asked. $\endgroup$
    – Matthew Conroy
    Commented Apr 21, 2017 at 21:50
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    $\begingroup$ Oh, well. That's true. I supposed that knowing $N(r)$ would imply knowing $r$. So sorry, my fault $\endgroup$
    – user3141592
    Commented Apr 21, 2017 at 21:56

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