On the Gauss sphere problem — Draconitic Research

Abstract

In this article we improve Heath-Brown's estimate for the error term in the three-dimensional sphere problem, proving #{x∈ℤ³:|x|≤ R}=4π/3R³+O_ε(R^55/42+ε). Incoporating the recent OpenAI quasi-GRH into our argument further improves this to #{x∈ℤ³:|x|≤ R}= 4π/3R³+O_ε(R^{2274811/1743872+ε}).

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BibTeX RIS

BibTeX

@misc{bellotti2026gauss,
  title = {{On the Gauss sphere problem}},
  author = {Chiara Bellotti and Bryce Kerr},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29636},
  eprint = {2610.29636},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29636},
}

RIS

TY  - UNPB
TI  - On the Gauss sphere problem
AU  - Chiara Bellotti
AU  - Bryce Kerr
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29636
UR  - https://arxiv.org/abs/2610.29636
AB  - In this article we improve Heath-Brown's estimate for the error term in the three-dimensional sphere problem, proving #{x∈ℤ^3:|x|≤ R}=4π/3R^3+O_ε(R^55/42+ε). Incoporating the recent OpenAI quasi-GRH into our argument further improves this to #{x∈ℤ^3:|x|≤ R}= 4π/3R^3+O_ε(R^2274811/1743872+ε).
ER  -

Abstract

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