Gauss circle problem

The Gauss circle problem asks how many integer lattice points lie inside a circle of radius r centered at the origin, and how far that count differs from πr².

Lattice point: A point whose coordinates are integers in a regularly spaced grid. The problem counts precisely these points inside a disk.
Carl Friedrich Gauss: German mathematician and physicist whose work shaped number theory, geometry, and many other fields. His treatment of the lattice count gave the problem its name.
Integer lattice: The set of points in Euclidean space whose coordinates are integers. The circle count is a two-dimensional count in this discrete grid.
Lattice-point discrepancy: The difference between the number of lattice points in a region and its expected volume-based approximation. The circle error is a canonical example of discrepancy for a growing region.
Divisor summatory function: The function that counts integer pairs whose product is at most a given number. Its summatory error is closely related to the circle-counting error through divisor sums.
Disquisitiones Arithmeticae: Gauss's 1801 book that systematized major parts of number theory. It contains Gauss's early estimate for the number of lattice points in a circle.
Euclidean disk: The set of points at distance at most a fixed radius from a center in Euclidean space. Its area supplies the leading approximation πr².
Gauss circle problem exponent: An exponent describing the strongest known power-law upper bound for the circle-counting error. Progress is measured by how far the error exponent falls below the elementary boundary scale.
Voronoi summation formula: A transformation formula relating sums of arithmetic functions to weighted sums involving oscillatory kernels. It turns divisor sums into oscillations that expose the circle error's structure.
Dirichlet divisor problem: The problem of estimating the error in counting integer pairs with product at most x. It became a closely related model for studying arithmetic counting errors.

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Legendre's three-square theoremRelated: Both problems study integer lattice points constrained by sums of squares.