A New Bound for the Gauss Circle Problem

Xi Cen, Xitao Gao, Cheng Zhang +1

Let $E(X)=\#\{(a,b)\in {\mathbb{Z}}^2:a^2+b^2\le X\}-\pi X$. We prove that $E(X)=O_{\epsilon }(X^{209/670+\epsilon })$ for every $\epsilon >0$, uniformly for real $X\ge 4$. This breaks the $5…