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arXiv · 2609.28253

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

Tristan Freiberg
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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.

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Tristan Freiberg. 2026-09-23. A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound. https://arxiv.org/abs/2609.28253

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