AI 中文总结
本文初等阐述倾斜筛法，证明每素数取一剩余类可覆盖长度为 x log x/((log_2 x)log_3 x) 的区间，改进 Erdős--Rankin 界，无需 Maynard 筛权重或超图覆盖定理。
AI 中文摘要
我们给出了 GPT-5.6~Sol 引入的倾斜筛法的一个初等阐述，证明对每个素数 $p \le x$ 取一个剩余类，可以覆盖长度为 \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x} \end{equation*} 的区间。这比经典的 Erdős--Rankin 界改进了一个因子 $(\log_{2} x)/(\log_{3} x)^{2}$。虽然弱于已知的最强结果，但它展示了倾斜及其相关的复合幸存者覆盖在没有 Maynard 筛权重或超图覆盖定理的情况下能实现什么。证明使用了素数定理、Mertens 倒数素数公式和初等概率。附录提供了历史综述和经典 Erdős--Rankin 界的自包含证明。
英文摘要
We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime $p \le x$ can cover an interval of length \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}. \end{equation*} This improves the classical Erdős--Rankin bound by a factor of $(\log_{2} x)/(\log_{3} x)^{2}$. Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem. The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability. The appendices provide a historical survey and self-contained proofs of the classical Erdős--Rankin bound.
Comments73 pages. Includes appendices on the classical Erdős--Rankin construction, the history of large prime gaps, and AI provenance. Edited prompt and response records supplied as ancillary files
