Cheng Zhang

A New Bound for the Gauss Circle Problem

Let $E(X)=\#\{(a,b)\in\Z^2:a^2+b^2\le X\}-\pi X$. 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:...