Number of lattice points on a sphere as $O(x^r)$

Question

It is known that the number of lattice points of $\mathbb{Z}^n\subset\mathbb{R}^n$ that lie inside a sphere of radius $R\in\mathbb{N}$ is asymptotic to $C_nR^{n}$, where $C_n$ is a constant depending only on the dimension $n$. I want to say that the number of lattice points that lie on insides a shell of thickness $1$ defined by $$(R+1)S^n-RS^n$$ is $O(R^{n-1})$ but I was not able to prove it and did not find a reference to this (presumed) fact. Essentially, I want to know if the number of lattice points inside the above mentioned shell can be given in big O notation. I will add that the motivation lies in calculating $$\sum_{a\in\mathbb{Z}^n,a\neq0}\frac{1}{||a||^s}$$ with $s>n$

Answer 1

With each $a \in \mathbb{Z}^n$ associate the half-open box $$B(a) := \prod_{\nu = 1}^n [a_{\nu}, a_{\nu} + 1).$$

If $a$ lies in the ball with radius $R$, then the box $B(a)$ is contained in the ball with radius $R + \sqrt{n}$, since $\operatorname{diam} B(a) = \sqrt{n}$. Each box has volume (Lebesgue measure) $1$, hence the number of points in the ball of radius $R$ is bounded by the volume of the ball with radius $R + \sqrt{n}$, which is

$$C_n (R+\sqrt{n})^n = C_n R^n + n^{3/2}C_n R^{n-1} + \dotsc = C_n R^n + O(R^{n-1}).$$

On the other hand, if $B(a)$ intersects the ball with radius $R - \sqrt{n}$, then $a$ lies in the ball with radius $R$. So the boxes associated with lattice points in the ball with radius $R$ cover the ball with radius $R - \sqrt{n}$, hence there must be at least $$C_n(R-\sqrt{n})^n = C_n R^n - n^{3/2}C_nR^{n-1} + \dotsc = C_nR^n + O(R^{n-1})$$

such boxes.

It follows that the number of lattice points in the ball with radius $R$ is $C_n R^n + O(R^{n-1})$, and thus the number of lattice points in the shell is

$$C_n(R+1)^n + O\bigl((R+1)^{n-1}\bigr) - C_nR^n + O(R^{n-1}) = O(R^{n-1}).$$

Answer 2

This is correct and you can find the constants. The volume of an $n-$ball is $C_nR^n$ with $C_n=\frac{\pi^{n/2}}{\sqrt \pi \Gamma((n+1)/2)}$ and the surface is $S_nR^{n-1}$ with $S_n=\frac {2\pi^{n/2}}{\Gamma(n/2)}$, which corresponds to the volume of a unit thickness shell. The lattice points are reasonably evenly distributed in these volumes.