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 $\epsilon >0$, $$N(t)=\pi t^2+O(t^{\theta +\epsilon }).$$ <a href="#CRM2023-def-theta">[CRM2023-def-theta]</a>

We define $$C_{64}:=\theta ,$$ the Gauss circle problem exponent.

The best established upper bound is due to Huxley: $$\theta \le \frac{131}{208}.$$ <a href="#CRM2023-ub-131-208">[CRM2023-ub-131-208]</a>

Hardy conjectured the optimal exponent $\theta =1/2$. <a href="#CRM2023-conj-1-2">[CRM2023-conj-1-2]</a>

The best established range currently is $$0\le C_{64}\le \frac{131}{208}.$$

Known upper bounds

Known lower bounds

Additional comments and links

• Conjectural value. Hardy's conjecture is $C_{64}=1/2$. <a href="#CRM2023-conj-1-2">[CRM2023-conj-1-2]</a>

• Reported but currently withdrawn source. <a href="#CRM2023">[CRM2023]</a> reports a later Bourgain-Watt claim $\theta \le 517/824+ϵ$. <a href="#CRM2023-ub-517-824">[CRM2023-ub-517-824]</a> The associated arXiv item is marked withdrawn. <a href="#BW2017-withdrawn">[BW2017-withdrawn]</a>

• Wikipedia page on Gauss's circle problem

References

• <a id="Hux2003"></a>[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

• <a id="CRM2023"></a>[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
• <a id="CRM2023-def-theta"></a>[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 $\theta$."
• <a id="CRM2023-ub-131-208"></a>[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)."
• <a id="CRM2023-ub-517-824"></a>[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 $\theta =517/824+ϵ$ for any $ϵ>0$."
• <a id="CRM2023-conj-1-2"></a>[CRM2023-conj-1-2] loc: PDF p.1, Introduction quote: "it is conjectured that the correct error is $|E(R)|=O(R^{1/2+ϵ})$ for any $ϵ>0$."

• <a id="BW2017"></a>[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
• <a id="BW2017-withdrawn"></a>[BW2017-withdrawn] loc: arXiv abstract page, header line below arXiv identifier quote: "This paper has been withdrawn by Nigel Watt"

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.