A note on the distribution of integer points on spheres

In this note, by applying the works of Böcherer and Schulze-Pillot, we evaluate asymptotically the variance of the Linnik distribution, and show as Y → ∞, $$\label{eq1} \sum _{n \leq Y,\, n \not \equiv 0 \, (\, {\rm mod} 4)} \frac{\sum _{|x|^{2} = n} P\left( \frac{x}{|x|} \right)}{n^{1/4}} {\overline{\frac{\sum _{|z|^{2} = n} Q \left(\frac{z}{|z|} \right)}{n^{1/4}}}} = \, c < P,\, Q > \, \Lambda (1/2,\, P) \, Y + o(Y) ,$$ for any spherical harmonics P, Q on S ² which are Hecke eigenforms, where | · | is the usual Euclidean norm on R ³, and c is an explicit constant.