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 $\sigma ({\Omega }_n){N_n}^{1+\epsilon }$, where $\sigma ({\Omega }_n)$ is the area of the ball ${\Omega }_n$ on the unit sphere, $N_n$ 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 $c\sigma ({\Omega }_n)N_n$, where $c$ is an absolute constant. This bound is of the conjectured order of magnitude.

arXiv comment48 pages