# Bound on 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.

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

• If $d\leq2$, is $\alpha$ still $d-2$? – Joonas Ilmavirta Oct 1 '14 at 12:08
• Chamizo and Iwaniec only state it as $V_dR^d+R^{d-2+\epsilon}$ for any $\epsilon>0$, and that it is proved for $d\ge4$. They also mention that the optimal exponents for $d=2$ and $d=3$ are conjectured to be $1/2+\epsilon$ and $1+\epsilon$, respectively, that the best upper bound proved for $d=2$ (at the time) is $46/73+\epsilon$, and the main result of the paper is a $29/22+\epsilon$ upper bound for $d=3$. – Emil Jeřábek Oct 1 '14 at 12:28
• @Joonas, I think you need $d \geq 4$. Anyway I'm interested in the case where $d$ is sufficiently large. Chamizo and Iwniec make the claim for $\alpha=d-2$ in the second paragraph in the paper I linked to. The say there is an elementary proof for it and cite the book by Fricker. This is the proof I'm interested in. – Guy Oct 1 '14 at 13:43
• Fricker’s book is easily found on the internet (with a choice between Springer and pirates), and does indeed contain a proof of the result, based on Jacobi's four-square theorem. It only holds in the form you stated it for $d\ge5$, whereas for $d=4$ one has $V_4R^4+O(R^2\log R)$. The result with the logs is due to Landau, the stronger form for $d\ge5$ apparently to Walfisz. There are several other references in the book, all German, if that’s the problem. – Emil Jeřábek Oct 1 '14 at 14:37
• German is indeed the problem... – Guy Oct 1 '14 at 16:24