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Level A · Machine-checkable Hard Number theory P-gauss-circle-problem

Gauss circle problem

It is conjectured that the correct bound is |E(r)| = O(r^1/2 + o(1)) [Ha59] Hardy, G. H. (1959). _Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work_(3rd ed.). New York: Chelsea Publishing Company. p. 67 See also https://arxiv.org/abs/2305.03549

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-gauss-circle-problem,
  title        = {Gauss circle problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/gauss-circle-problem}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The question

It is conjectured that the correct bound is

[Ha59] Hardy, G. H. (1959). _Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work_(3rd ed.). New York: Chelsea Publishing Company. p. 67

See also https://arxiv.org/abs/2305.03549

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.GaussCircleProblem.

theorem error_isBigO : ∃ (o : ℝ → ℝ) (_ : Tendsto o atTop (𝓝 0)),
    E =O[atTop] fun r => r ^ (1/2 + o r)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.