OpenFrontiers
Join
← Back to the index
OP-12079OpenNumber theory

Gauss circle problem exponent (C64C_{64})

Gauss circle problem exponent: determine C64C_{64} or improve its bounds; the repository records 0 (lower) and 131208\frac{131}{208} (upper).

MSC 11P21 · Lattice points in specified regions

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let

N(t) := #{(m,n)∈Z2: m2+n2≤t2}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 tt centered at the origin. The Gauss circle problem is to find the smallest exponent θ\theta such that, for every ε>0\varepsilon>0,

N(t) = πt2 + O(tθ+ε).N(t)\ =\ \pi t^2\ +\ O(t^{\theta+\varepsilon}).

[CRM2023-def-theta]

We define

C64 := θ,C_{64}\ :=\ \theta,

the Gauss circle problem exponent.

The best established upper bound is due to Huxley:

θ ≤ 131208.\theta\ \le\ \frac{131}{208}.

[CRM2023-ub-131-208]

Hardy conjectured the optimal exponent θ=1/2\theta=1/2. [CRM2023-conj-1-2]

The best established range currently is

0 ≤ C64 ≤ 131208.0\ \le\ C_{64}\ \le\ \frac{131}{208}.

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

References (from the repository page)

  • [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≥0R\ge 0 in R2\mathbb{R}^2 ... the error being initially in the form of O(Rθ)O(R^{\theta}), mathematicians have been trying to minimise θ\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≈0.6298131/208\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 θ=517/824+ϵ\theta = 517/824 + \epsilon for any ϵ>0\epsilon > 0."

  • [CRM2023-conj-1-2] loc: PDF p.1, Introduction quote: "it is conjectured that the correct error is ∣E(R)∣=O(R1/2+ϵ)|E(R)| = O(R^{1/2+\epsilon}) for any ϵ>0\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

  • [BW2017-withdrawn] loc: arXiv abstract page, header line below arXiv identifier quote: "This paper has been withdrawn by Nigel Watt"

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound 131208\frac{131}{208}. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
11 Trivial bound N(t)=πt2+O(t)N(t)=\pi t^2+O(t).
131/208131/208 [Hux2003] Huxley's bound (long-standing record). [CRM2023-ub-131-208]

Known lower bounds

Bound Reference Comments
00 Trivial from the definition C64≥0C_{64}\ge 0.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Gauss circle problem exponent ($C_{64}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.