On the Gauss sphere problem

Chiara Bellotti; Bryce Kerr

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arXiv · 2610.29636

On the Gauss sphere problem

Chiara Bellotti·Bryce Kerr

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Abstract

In this article we improve Heath-Brown's estimate for the error term in the three-dimensional sphere problem, proving \[ \#\{\mathbf x\in\mathbb Z^3:\|\mathbf x\|\le R\}=\frac{4π}{3}R^3+O_\varepsilon(R^{55/42+\varepsilon}). \] Incoporating the recent OpenAI quasi-GRH into our argument further improves this to \[ \#\{\mathbf x\in\mathbb Z^3:\|\mathbf x\|\le R\}= \frac{4π}{3}R^3+O_\varepsilon(R^{\frac{2274811}{1743872}+\varepsilon}). \]

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Chiara Bellotti, Bryce Kerr. 2026-10-21. On the Gauss sphere problem. https://arxiv.org/abs/2610.29636

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