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arXiv · 2610.21730

A New Bound for the Gauss Circle Problem

Xi Cen·Xitao Gao·Cheng Zhang·Junyong Zhang
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Abstract

Let $E(X)=\#\{(a,b)\in\Z^2:a^2+b^2\le X\}-πX$. We prove that $E(X)=O_\varepsilon(X^{209/670+\varepsilon})$ for every $\varepsilon>0$, uniformly for real $X\ge4$. 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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Xi Cen, Xitao Gao, Cheng Zhang, Junyong Zhang. 2026-10-15. A New Bound for the Gauss Circle Problem. https://arxiv.org/abs/2610.21730

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