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OpenNumber Theory·1837

Gauss Circle Problem

Estimate how many integer lattice points lie inside a circle of radius r.

Formula

Lattice point count
N(R)=#{(m,n)∈Z2:m2+n2≤R2}=πR2+E(R)N(R)=\#\{(m,n)\in\mathbb{Z}^2: m^2+n^2\le R^2\}=\pi R^2+E(R)

Count lattice points inside a circle of radius R; the error E(R) is the central object of study.

Hardy conjecture
E(R)=O(R1/2+ε)∀ ε>0E(R)=O\bigl(R^{1/2+\varepsilon}\bigr)\quad\forall\,\varepsilon>0

Hardy and Landau conjectured the optimal error exponent is 1/2 — the trivial bound is O(R) and the best known is 131/208.

Best known bound (Huxley 2000)
E(R)=O(R131/208),131208≈0.6298E(R)=O\bigl(R^{131/208}\bigr),\qquad \tfrac{131}{208}\approx 0.6298

Huxley's exponential-sum estimate gives the best unconditional bound, still far from the conjectured exponent 1/2.

Summary

The Gauss circle problem asks for the sharp error term when counting lattice points in a disk. The image is simple: the main term is area, and the hard part lives near the boundary.

Sources

On the expression of a number as the sum of two squares

1915

Exponential sums and lattice points III

2003

Improvement on Gauss circle Problem and Dirichlet divisor Problem

arXiv:2308.148592023

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