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.
(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 $N(r)$.