Rational Points on a Sphere

Say we want to count rational points on the unit sphere of a bounded height and study their distribution.

Counting:

The number of rational points on the sphere of height bounded by {T} is given by {\frac{3}{2 \kappa} T^{2}} where {\kappa = L(2, \chi_{-4}) =\frac{1}{1^2} -\frac{1}{3^2} + \frac{1}{5^2}- \frac{1}{7^2} +\cdots \approx 0.9159}

The rational points of height {n} correspond to primitive integer points on the sphere {\{ x^2+y^2+z^2 =n^2\}} of radius {n} by clearing the denominators.

If {s(n)} is the number of points of height {n}, {r(m)} number of points on sphere {\{ x^2+y^2+z^2 =m\}}of radius {\sqrt{m}}, we have

\displaystyle r(n^2) = \sum_{d|n} s\left(\frac{n}{d}\right)

Here {s(n)} only counts the primitive points and you sum over al the divisors to account for all points.

So we have

\displaystyle \sum_{n=1}^{\infty}\frac{ r\left(n^{2}, 1\right)}{n^{s}}= \zeta(s)\cdot \sum_{n=1}^{\infty}\frac{ s(n)}{n^{s}}

Using the formulae that relate {r(n)} to the class number {h(-4n)} (Gauss), we get the following identity:

\displaystyle \sum_{n=1}^{\infty}\frac{ r\left(n^{2}, 1\right)}{n^{s}}=6\left(1-2^{1-s}\right) \frac{\zeta(s) \zeta(s-1)}{L\left(s, \chi_{-4}\right)}

Thus we see that

\displaystyle \sum_{n=1}^{\infty}\frac{ s(n)}{n^{s}}= 6\left(1-2^{1-s}\right) \frac{\zeta(s-1)}{L\left(s, \chi_{-4}\right)}

Perron’s formula or some Tauberian argument gives

\displaystyle \sum_{n\le T} s(n) \sim \frac{3}{2 \kappa} T^{2}

Equidistribution: We want to study the equidstribution of the points we need to prove that for any continuous function on the sphere

\displaystyle \frac{\displaystyle \sum_{n \le T} s(n, f)}{ \displaystyle \sum_{n \le T} s(n, 1)} \sim \int_{S^{2}} f(\sigma) d \sigma

where

\displaystyle s(n, f) = \sum_{\substack{x^2+y^2+z^2=1\\ H(x,y,z)=n}} f(x,y,z)

In fact, it is enough to prove it for harmonic polynomials {P_d} on the sphere.

Recall, that while counting ie., for {f=1}, we used the information about explicit formulae for {r(n).}
We had an identity relating the Dirichlet series for {r(n^2)} in terms of other known L-functions.

For {s(n, P_d)}, we need information about {r(n^, P_d)}, where

\displaystyle r(n, P_d) = \sum_{x^2+y^2+z^2=n} P_d( \frac{x}{\sqrt n}, \frac{y}{\sqrt n},\frac{z}{\sqrt n})

The required information is provided by Shimura Lift:

\displaystyle \sum_{n=1}^{\infty}\frac{ s(n, P_d)}{n^{s}}=\frac{L\left(s-\frac{1}{2}, F\right)}{\zeta(s) L\left(s, \chi_{-4}\right)}

where {F} is a cusp form on {\Gamma_0(2)} of weight {2d+2}

{L(s, F)} converges absolutely for {\Re(s) > 1} (The bound on the coefficients can be obtained by Rankin-Selberg for instance)

Thus we see have that

\displaystyle \displaystyle \sum_{n \le T} s(n, P_d) \ll T^{3/2}

\displaystyle \frac{\displaystyle \sum_{n \le T} s(n, f)}{ \displaystyle \sum_{n \le T} s(n, 1)} \rightarrow 0

If you want to equidistribution of points of a fixed height, you need to use the above Shimura relation and a bound better than Hecke bound on the Fourier coefficients of {F.}

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