Gauss circle problem exponent
Description of constant
Let \(N(t)\ :=\ \#\{(m,n)\in\mathbb{Z}^2:\ m^2+n^2\le t^2\}\) be the number of integer lattice points inside the (closed) disk of radius $t$ centered at the origin. The Gauss circle problem is to find the smallest exponent $\theta$ such that, for every $\varepsilon>0$, \(N(t)\ =\ \pi t^2\ +\ O(t^{\theta+\varepsilon}).\) [CRM2023-def-theta]
We define \(C_{64}\ :=\ \theta,\) the Gauss circle problem exponent.
The best established upper bound is due to Huxley: \(\theta\ \le\ \frac{131}{208}.\) [CRM2023-ub-131-208]
Hardy conjectured the optimal exponent $\theta=1/2$. [CRM2023-conj-1-2]
The best established range currently is \(0\ \le\ C_{64}\ \le\ \frac{131}{208}.\)
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial bound $N(t)=\pi t^2+O(t)$. | |
| $131/208$ | [Hux2003] | Huxley’s bound (long-standing record). [CRM2023-ub-131-208] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial from the definition $C_{64}\ge 0$. |
Additional comments and links
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Conjectural value. Hardy’s conjecture is $C_{64}=1/2$. [CRM2023-conj-1-2]
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Reported but currently withdrawn source. [CRM2023] reports a later Bourgain-Watt claim $\theta\le 517/824+\epsilon$. [CRM2023-ub-517-824] The associated arXiv item is marked withdrawn. [BW2017-withdrawn]
References
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[Hux2003] Huxley, M. N. Exponential sums and lattice points III. Proceedings of the London Mathematical Society (3) 87 (2003), no. 3, 591-609. DOI: https://doi.org/10.1112/S0024611503014485. Google Scholar
- [CRM2023] Costa, Krits; Ruiz Martínez, Jon. Report on Presentation: Equidistribution and the Gauss Circle Problem. ETH Zurich course report (2023). PDF: https://metaphor.ethz.ch/x/2023/hs/401-3100-73L/ex/gausscircleproblem.pdf. Google Scholar
- [CRM2023-def-theta] loc: PDF p.1, Introduction quote: “Given a circle of radius $R\ge 0$ in $\mathbb{R}^2$ … the error being initially in the form of $O(R^{\theta})$, mathematicians have been trying to minimise $\theta$.”
- [CRM2023-ub-131-208] loc: PDF p.1, Introduction quote: “the period ending with british Martin Neil Huxley finding the best known bound until then, $131/208\approx 0.6298$ (2000).”
- [CRM2023-ub-517-824] loc: PDF p.1, Introduction quote: “the best improvement on the upper bound we have today is still "far off" this result and attributed to belgian Jean Bourgain and English Nigel Watt … they found in 2017 that $\theta = 517/824 + \epsilon$ for any $\epsilon > 0$.”
- [CRM2023-conj-1-2] loc: PDF p.1, Introduction quote: “it is conjectured that the correct error is $|E(R)| = O(R^{1/2+\epsilon})$ for any $\epsilon > 0$.”
- [BW2017] Bourgain, Jean; Watt, Nigel. Mean square of zeta function, circle problem and divisor problem revisited. arXiv:1709.04340 (2017; revised 2023). arXiv page: https://arxiv.org/abs/1709.04340. Google Scholar
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.