PreprintCC BY 4.0arXiv
A New Bound for the Gauss Circle Problem
Abstract
Let E(X)=#{(a,b)∈Z2:a2+b2≤ X}−πX. We prove that E(X)=Oε(X209/670+ε) for every ε>0, uniformly for real X≥4. This breaks the 5/16 threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements: a cone distribution estimate that preserves separate fourth- and sixth-power contributions in the double large sieve, and an explicit inversion of Taylor-coefficient constraints that cancels the linear error in a pair-counting argument. These refinements, together with the Robert--Sargos and Watt spacing estimates and the Vinogradov mean value theorem, yield the improvement.
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BibTeX
@misc{cen2026bound,
title = {{A New Bound for the Gauss Circle Problem}},
author = {Xi Cen and Xitao Gao and Cheng Zhang and Junyong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21730},
eprint = {2610.21730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21730},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A New Bound for the Gauss Circle Problem
AU - Xi Cen
AU - Xitao Gao
AU - Cheng Zhang
AU - Junyong Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21730
UR - https://arxiv.org/abs/2610.21730
AB - Let E(X)=#{(a,b)∈Z^2:a^2+b^2≤ X}−πX. We prove that E(X)=O_ε(X^209/670+ε) for every ε>0, uniformly for real X≥4. This breaks the 5/16 threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements: a cone distribution estimate that preserves separate fourth- and sixth-power contributions in the double large sieve, and an explicit inversion of Taylor-coefficient constraints that cancels the linear error in a pair-counting argument. These refinements, together with the Robert--Sargos and Watt spacing estimates and the Vinogradov mean value theorem, yield the improvement.
ER -