Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Cen, Xi, Gao, Xitao, Zhang, Cheng, Zhang, Junyong
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G7293072536, G8054159703
OpenAlex ID: W7222733598