A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound

Abstract

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.

73 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