Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles

Abstract

Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-$R$ disc by its area is $O\left(R^{1/2+o\left(1\right)}\right)$. One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.