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.
(This is a repost of a question I asked in the Mathematics Stack Exchange)