Variance Estimates in Linnik’s Problem

Abstract

We evaluate the variance of the number of lattice points in a small randomly rotated spherical ball on a surface of 3-dimensional sphere centered at the origin. Previously, Bourgain, Rudnick, and Sarnak showed conditionally on the Generalized Lindelöf Hypothesis that the variance is bounded from above by,σ (Ω)Nₙ1+ϵ, where σ (Ω) is the area of the ball Ω on the unit sphere and is Nn is the total number of solutions of Diophantine equation x²+y²+z²=n. Assuming the Grand Riemann Hypothesis and using the moments method of Soundararajan and Harper, we establish the upper bound of the form,cσ(Ω)Nₙ where c is an absolute constant. This bound is of the conjectured order of magnitude.