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 is given by where
The rational points of height correspond to primitive integer points on the sphere of radius by clearing the denominators.
If is the number of points of height , number of points on sphere of radius , we have
Here only counts the primitive points and you sum over al the divisors to account for all points.
So we have
Using the formulae that relate to the class number (Gauss), we get the following identity:
Thus we see that
Perron’s formula or some Tauberian argument gives
Equidistribution: We want to study the equidstribution of the points we need to prove that for any continuous function on the sphere
where
In fact, it is enough to prove it for harmonic polynomials on the sphere.
Recall, that while counting ie., for , we used the information about explicit formulae for
We had an identity relating the Dirichlet series for in terms of other known L-functions.
For , we need information about , where
The required information is provided by Shimura Lift:
where is a cusp form on of weight
converges absolutely for (The bound on the coefficients can be obtained by Rankin-Selberg for instance)
Thus we see have that
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