OEIS A00023: Sphere lattice-point error in a 3D ball

In this episode we explore the classic problem of counting lattice points inside a three-dimensional sphere and how that count deviates from the smooth volume. Define A_N as the number of integer triples i, j, k with i^2 + j^2 + k^2 ≤ N, and compare it to the sphere’s volume V_N. The difference P_N = A_N − V_N is the lattice-point error term. We’ll connect this to the 2D Gauss circle problem, discuss what A00092 reveals about record-high errors, and outline the analytic tools that drive current bounds—harmonic analysis, exponential sums, and smoothing tricks. We’ll also glimpse higher-dimensional generalizations and what these fluctuations say about how a discrete grid fills continuous space.

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