Variance estimates in Linnik's problem

TL;DR

This paper proves an upper bound on the variance of lattice points in small spherical caps on a 3D sphere, assuming the Grand Riemann Hypothesis, matching the conjectured order of magnitude.

Contribution

It establishes a new upper bound on the variance under GRH, improving previous conditional results and confirming the conjectured order of magnitude.

Findings

Upper bound of variance proportional to $\sigma(\Omega_n) N_n$

Assumption of GRH enables the proof

Results match the conjectured order of magnitude

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 Lindel\"of Hypothesis that the variance is bounded from above by $\sigma(\Omega_n){N_n}^{1+\varepsilon}$, 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.