Variance Estimates in Linnik’s Problem

Abstract

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 $\sigma (\Omega ){N_n}^{1+\varepsilon }$, where $\sigma (\Omega )$ is the area of the ball $\Omega $ on the unit sphere and $N_n$ is the total number of solutions of Diophantine equation $x^2 + y^2 + z^2 = n$. Assuming the Grand Riemann Hypothesis and using the moments method of Soundararajan and Harper, we establish the upper bound of the form $c\sigma (\Omega ) N_n$, where $c$ is an absolute constant. This bound is of the conjectured order of magnitude.