On the Gauss sphere problem

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