Open problem, Arithmetic and number theory, Analysis, posed 1834

Gauss circle problem

Open

Let $N(r) = \#\{(m,n) \in \mathbb{Z}^2 : m^2 + n^2 \le r^2\}$ be the number of integer lattice points inside a circle of radius $r$ centered at the origin, and write $N(r) = \pi r^2 + E(r)$ . Determine the infimum $\theta$ of all exponents such that $E(r) = O(r^{\theta+\varepsilon})$ as $r \to \infty$ for every $\varepsilon > 0$ ; it is conjectured that $\theta = \tfrac{1}{2}$ .

Posed by, 1834Carl Friedrich Gauss

Research frontier as of 2026

As of 2026, the Gauss circle problem remains open, with the true exponent $\theta$ known to lie in $[\tfrac{1}{2}, \tfrac{131}{208}]$ . On the upper-bound side, Martin Huxley's 2003 peer-reviewed bound $\theta \le \tfrac{131}{208} \approx 0.62981$ (after a 2017 decoupling preprint of Bourgain and Watt claiming $\tfrac{517}{824} \approx 0.62743$ was withdrawn in 2023, with subsequent decoupling preprints such as Li–Yang 2023 continuing this direction) represents the benchmark of exponential-sum technology. On the lower-bound and probabilistic side, Heath-Brown (1999) proved that $r^{-1/2}E(r)$ has a non-Gaussian limiting distribution, and Soundararajan (2003) improved the extreme oscillation bound to $E(r) = \Omega\!\left(r^{1/2}(\log r)^{1/4}(\log\log r)^{\frac{3}{4}(2^{4/3}-1)}(\log\log\log r)^{-5/8}\right)$ .

Best known results

Published upper bound: $E(r) = O\!\left(r^{131/208}(\log r)^{18627/8320}\right)$ , so $\theta \le \tfrac{131}{208} \approx 0.62981$ (Huxley, 2003).
Mean-square asymptotic: $\int_1^T |E(r)|^2\,dr = c T^2 + O(T^{3/2+\varepsilon})$ , which shows that $|E(r)|$ is $O(r^{1/2})$ on average.
Oscillation lower bound: $E(r) = \Omega\!\left(r^{1/2}(\log r)^{1/4}(\log\log r)^{\frac{3}{4}(2^{4/3}-1)}(\log\log\log r)^{-5/8}\right)$ (Soundararajan, 2003).

Tools and where they stop

Tool	Achieved	Where it stops
Hardy–Voronoi Bessel-series identity and Poisson summation	Expands $E(r)$ into a series of Bessel functions $r \sum_{n=1}^{\infty} r_2(n) n^{-1/2} J_1(2\pi r\sqrt{n})$ , converting lattice-point counting into oscillatory exponential sums.	Truncating the Voronoi series at length $N$ leaves a smoothing error of $O(r N^{-1/2})$ ; reaching $\theta = \tfrac{1}{2}$ would require square-root cancellation all the way up to $N \approx r$ .
Discrete Hardy–Littlewood method (Bombieri–Iwaniec–Mozzochi, Huxley)	Applies rational approximation of the phase and large-sieve resonance bounds to push the exponent down to $\theta \le \tfrac{131}{208}$ .	Relies on local polynomial approximations of the phase on short intervals, which cannot detect global arithmetic independence of the frequencies $\sqrt{n}$ .

Open questions

Does $E(r) = O(r^{1/2+\varepsilon})$ hold for every $\varepsilon > 0$ ?
What is the exact power of $\log r$ in the maximal order of $|E(r)| / r^{1/2}$ as $r \to \infty$ ?

References

G. H. Hardy (1915). On the expression of a number as the sum of two squares
Henryk Iwaniec, C. J. Mozzochi (1988). On the divisor and circle problems · DOI:10.1016/0022-314X(88)90049-7
Martin N. Huxley (2003). Exponential sums and lattice points III · DOI:10.1112/S0024611503014485