# Counting Lattice Points in an $n$-sphere and Sphere Packing

Does knowing the exact number of $\mathbb{Z}^{n}$-lattice points in an $n$-sphere of arbitrary radius help one compute $n$-sphere packing density and kissing number?

Thanks!

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I have no idea as to whether it helps at all, but even if it would, I wanted to point out that the number of lattice points $l(r)$ in a sphere of radius $r$ isn't known more than asymptotically, even for $n = 2$. More precisely for $n =2$, it is known that $l(r) - \pi r = O(r^{\alpha})$ for some $0.25 \lt \alpha \lt 3.2$ (the best known upper bound is that $\alpha = \frac{23}{73}+\varepsilon \approx 0.3151$ works - Huxley 1993. The lower bound $0.25$ is due to Landau and Hardy). See the first few pages of P. de la Harpe, Topics in geometric group theory for a number of references. – t.b. Feb 21 '11 at 10:53
@Theo: Not sure exactly what result you're citing? What is $l(r)$, which you have approaching $\pi r$? At any rate, Mathworld ("Gauss's Circle Problem") has the number of lattice points in the circle of radius $r$ with center at the origin as $\pi r^2 + O(r^\alpha)$ with $\alpha=46/73$ (Huxley 1993), and more recently $\alpha=131/208$ (Huxley 2003). – mjqxxxx Feb 21 '11 at 13:29
@mjqxxxx: I think I cite exactly the same result. My mistake, I should have written "in the circle of radius $r^2$" instead of $r$, so $l(r) = \{(a,b) \in \mathbb{Z}^{2}\,:\,a^2 + b^2 \leq r\}$. – t.b. Feb 21 '11 at 13:33
All good comments. Thanks, everyone. However, my question still stands. Suppose one did have a method of computing the exact number of lattice points in an $n$-sphere. Would it help in understand sphere packing, at least in lattices unimodularly equivalent to $\mathbb{Z}^n$? – user02138 Feb 25 '11 at 3:48