OEIS A000323: Gauss circle problem—record lattice-point errors

In this Deep Dive, we explore A000323, the sequence of n at which the Gauss circle error term sets a new record. We define A(n) as the number of integer pairs (i,j) with i^2 + j^2 ≤ n and P(n) = A(n) − πn, the gap between lattice-point counts and the circle’s area. A000323 lists A(n) only at the points where |P(n)| hits a fresh all-time maximum. We trace the history: Hardy’s Ω(n^{1/4}) lower bound, Chen’s O(n^{0.324…}) upper bound, and the conjecture that the true rate is n^{1/4+ε}. We revisit Mitchell’s 1966 IBM 7094 computations up to n = 250,000, which hinted a negative-bias in extreme errors and aligned more with Chen’s bound than the simpler guess. The episode highlights the rich interplay between geometry and number theory and why the Gauss circle problem remains open.

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