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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 ≤ x can cover an interval of length ≫ (xlog x)/((log2 x)log3 x). This improves the classical Erdős--Rankin bound by a factor of (log2 x)/(log3 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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Cite

BibTeXRIS
BibTeX
@misc{freiberg2026tyros,
  title = {{A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound}},
  author = {Tristan Freiberg},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28253},
  eprint = {2609.28253},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28253},
}
RIS
TY  - UNPB
TI  - A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound
AU  - Tristan Freiberg
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28253
UR  - https://arxiv.org/abs/2609.28253
AB  - 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 ≤ x can cover an interval of length ≫ (xlog x)/((log_2 x)log_3 x). 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.
ER  -