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Products of two elements of Granville's random model in residue classes

  • William D. Banks
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Abstract

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo q contains a product of two primes not exceeding q; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range qΘ for any fixed Θ> 3/4, in Granville's random model of the primes. Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes.

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Cite

BibTeXRIS
BibTeX
@misc{banks2026products,
  title = {{Products of two elements of Granville's random model in residue classes}},
  author = {William D. Banks},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37434},
  eprint = {2609.37434},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37434},
}
RIS
TY  - UNPB
TI  - Products of two elements of Granville's random model in residue classes
AU  - William D. Banks
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37434
UR  - https://arxiv.org/abs/2609.37434
AB  - Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo q contains a product of two primes not exceeding q; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range q^Θ for any fixed Θ> 3/4, in Granville's random model of the primes. Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes.
ER  -