# Counting lattice points inside a three-dimensional ellipsoid

I want to answer the following simple question:

Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in $\mathbb{Z}^3$ are inside the ellipsoid?

The answer is given by the volume of the ellipsoid (of order $Z^{3/2}$) plus an error term, and I am interested in the strongest possible error term, which I hope would be $O(Z^{3/4})$ or better. The error term should also depend on $Q$ in some fashion which is explicitly described.

It seems like this question must have been extensively studied by now. But I was unable to find a suitable reference. Two references that approach what I'm looking for are:

1. Schmidt (Lemma 1, Northcott's theorem on Heights II. The quadratic case, Acta Arith., 1995) proves that if $S$ is (for example) a convex body in $n$ dimensions, lying in a ball of radius $r$, and $\Lambda$ is any lattice of rank $r$, the number of lattice points of $\Lambda$ inside $S$ is $$\frac{\rm{Vol}(S)}{\det(\Lambda)} + O\left(\frac{\lambda_n r^{n - 1}}{\det(\Lambda)}\right),$$ where $\lambda_n$ is the largest successive minimum of $\Lambda$. After a change of variable this is equivalent to my question, and the answer is of the form I am looking for -- but I believe a better error term should be possible when $S$ is an ellipsoid.

2. Bentkus and Götze (main theorem, On the lattice point problem for ellipsoids, Acta Arith., 1997) formulate the question in the same way that I did, and obtain a power savings of $Z$ in the error term (as opposed to $Z^{1/2}$, which is what can be deduced from Schmidt's paper or any similar geometry of numbers method). Writing the quadratic form as $\langle Qx, x \rangle$, the theorem specificies the dependence of the error term on the eigenvalues of $Q$. This is of the shape that I'm interested in, but the paper requires that the dimension be at least 9.

I skimmed through the likelier looking references in the latter paper, as well as the books on counting lattice points of Krätzel and Fricker, and I found nothing. The problem of counting lattice points in three-dimensional ellipsoids is addressed, but if the dependence of the error term on the ellipsoid is made explicit, then I missed it.

Finally I should mention that I know how to solve my own problem: write down the Epstein zeta function associated to $Q$, and estimate its partial sums using Perron's formula and the method of Landau and Chandrasekharan-Narasimhan. (The methods in the books above don't use Epstein zeta functions, but after a brief reading they seem like equivalent arguments that don't go through the usual zeta function machinery.) The dependence of the error term on $Q$ can be described in terms of the functional equation for the Epstein zeta function.

But I would prefer to avoid inventing the wheel if I can help it. Does anyone know if such a theorem has already been proved? Thank you very much.

• suspect this is not what you want, however: zakuski.math.utsa.edu/~kap/Duke_Schulze_Pillot_1990.pdf Feb 28, 2016 at 20:46
• Not what I'm looking for, but definitely interesting! (Indeed, their result seems much harder than what I'm looking for.) Feb 28, 2016 at 20:58
• Just checking, does your quadratic form have integer coefficients? Feb 28, 2016 at 20:59
• Frank, I think this might reveal something about the minimum size of the error term. Do your estimates as carefully as you can for the two forms $x^2 + 3 y^2 + 36 z^2$ and $3 x^2 + 4 y^2 + 9 z^2.$ The splitting integers for the genus are squares, for other positive integers the number of lattice points on the ellipsoid come out the same. The spinor exceptional integers are $w^2,$ where all prime factors $p$ of $w$ have $p \equiv 1 \pmod 3.$ My guess is that this shows the worst possible comparison for two forms with equal determinants. Feb 28, 2016 at 22:41
• @GerryMyerson: True, nobody can get the conjectured error term. But where "naive geometry of numbers" gets you an error term of order $X^{1/2}$, you can smooth the characteristic function of the circle, use Poisson summation, nontrivially analyze the resulting Bessel functions, and get an error term of $X^{1/3}$. (The Epstein zeta function machinery is roughly equivalent to this.) You can save a little bit more with effort, but I'd be very happy with an analogue of this $X^{1/3}$. Feb 29, 2016 at 13:40