The number of lattice points in d-dimensional ball The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-dimensional ball.
The paper cites a German book Einführung in die Gitterpunktlehre by F. Fricker for the proof. A similar statement also appears in this paper, again without proof (or citation).
References to the proof of the above statement will be very helpful.

The second linked paper has several citations. See also the intro in this paper: Fernando Chamizo and Carlos Pastor: Lattice points in bodies of revolution II.
