Conjectured one-quarter-plus-epsilon error bound in the Gauss circle problem

Establish the conjectured error bound for the Gauss circle problem by proving that, for x > 0, the discrepancy |C(√x) − πx| is O(x^(1/4+ε)) for every ε > 0, where C(√x) denotes the number of integer lattice points inside the circle of radius √x centered at the origin.

Background

The Gauss circle problem studies the discrepancy between the lattice-point count C(r) in a circle of radius r and the area πr^2. Writing x = r^2, the error term is commonly expressed as E(x) = C(√x) − πx. Over more than a century, successive improvements have lowered the exponent in bounds of the form E(x) = O(x^α), culminating in results such as O(x^37/112).

A long-standing conjecture predicts the optimal bound E(x) = O(x^1/4+ε) for any ε > 0. The paper notes that Iwaniec and Mozzochi related this conjectured bound to the Riemann Hypothesis, underscoring its centrality and difficulty.