Math Conjectures

Gauss Circle-Problem Exponent Conjecture

OPENMajorExact constant problemProposed c. 1915 · Standard version

Canonical statement

For R≥1R\ge1, let
P(R)=#{(m,n)∈Z2:m2+n2≤R2}−πR2. P(R)=\#\{(m,n)\in\mathbb Z^2:m^2+n^2\le R^2\}-\pi R^2.
Then for every ε>0\varepsilon>0,
P(R)=Oε(R1/2+ε)(R→∞). P(R)=O_\varepsilon(R^{1/2+\varepsilon}) \quad(R\to\infty).
View source LaTeX
For \(R\ge1\), let
\[
  P(R)=\#\{(m,n)\in\mathbb Z^2:m^2+n^2\le R^2\}-\pi R^2.
\] Then for every \(\varepsilon>0\),
\[
  P(R)=O_\varepsilon(R^{1/2+\varepsilon})
  \quad(R\to\infty).
\]
NT-LATTICE-027:[Hardy1915Circle][Huxley2003Lattice][BourgainWatt2017]
Number theoryDiscrete geometry

Notes

Write P(R)P(R) for the number of integer lattice points in the disc of radius RR minus its area πR2\pi R^2. The Gauss circle problem, in its modern sharp form, conjectures that P(R)=Oε(R1/2+ε)P(R)=O_\varepsilon(R^{1/2+\varepsilon}) for every ε>0\varepsilon>0. Gauss initiated the counting problem much earlier, with the elementary bound P(R)=O(R)P(R)=O(R); the precise exponent conjecture took shape around 1915 in the work of Hardy [Hardy1915Circle].

Hardy showed that P(R)P(R) genuinely fluctuates on the square-root scale, so the exponent 1/21/2 cannot be lowered and the conjecture is optimal apart from the RεR^\varepsilon factor [Hardy1915Circle]. On the upper-bound side, a century of refinements of exponential-sum methods has pushed the admissible exponent steadily downward, with Huxley's estimates [Huxley2003Lattice] and the subsequent improvement of Bourgain and Watt [BourgainWatt2017] holding the strongest results; all proved exponents remain strictly greater than 1/21/2. The conjecture is thus open precisely in the range between the best current exponent and 1/21/2, and closing that range appears to require ideas beyond present exponential-sum technology.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.