Spatial statistics for lattice points on the sphere I‎: ‎Individual results

Abstract

‎We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares‎. ‎We examine several statistics of these point sets‎, ‎such as the electrostatic potential‎, ‎Ripley's function‎, ‎the variance of the number of points in random spherical caps‎, ‎and the covering radius‎. ‎Some of the results are conditional on the Generalized Riemann Hypothesis‎. ‎