OEIS A000328: Circle problem — lattice points inside a circle

We dive into A000328, the Gaussian circle problem: how many integer lattice points (x, y) lie inside or on a circle of radius n. Start with the main term a(n) ~ πn^2 and the elusive remainder r(n) = a(n) − πn^2. We trace the historical bounds — Hardy and Landau showed the lower limit Ω(n^{1/2}); over the decades mathematicians sharpened the upper bound, with the current best known result due to Huxley giving a(n) − πn^2 = O(n^{131/208}) ≈ O(n^{0.6298}). We also connect to the sum-of-two-squares function and the broader circle problem in higher dimensions. This episode highlights how a simple counting question reveals deep links between geometry, number theory, and analysis, and why the quest to pin down the exact size of the error term remains an active area of research in the OEIS and beyond.

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