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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.

Batominovski
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Guy
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    I think you need $d\ge 4$. The statement is definitely false for $d=1$ and $d=2$. – deinst Sep 16 '14 at 18:01
  • You're probably right. I'm only interested in cases where $d$ is large enough. – Guy Sep 19 '14 at 15:26
  • While quite late to the question, this answer seems quite relevant. – Mark Schultz-Wu Feb 25 '19 at 23:42
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    6 years late. I don't know about this $\alpha=d-2$. If other people are interested and can settle for $\alpha=d-1$, then a complete proof appears in Reed-Simon Volume 4, proof of Proposition 2, page 267 (it's related to eigenvalue problems). – Lonewolf Aug 21 '21 at 11:16
  • For the people who are not able to open the first paper link (seems it's not working now) an alternate link is here: http://dmle.icmat.es/pdf/MATEMATICAIBEROAMERICANA_1995_11_02_08.pdf – NumDio Jul 12 '23 at 09:57

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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.

domotorp
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