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Hardy conjectured that the error term arising from approximating the numberof lattice points lying in a radius-$R$ disc by its area is $O(R^{1/2+o(1)})$.One source of support for this conjecture is a folklore heuristic that usesi.i.d. random variables to model the lattice points lying near the boundary andsquare-root cancellation of sums of these random variables. We examine thisheuristic by studying how these lattice points interact with one another andprove that their autocorrelation is determined in terms of a random model.Additionally, it is shown that lattice points near the boundary which are "wellseparated" behave independently. We also formulate a conjecture concerning thedistribution of pairs of these lattice points.

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