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Clusters of primes revisited

  • Paweł Lewulis
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Abstract

We revisit the generalized Rayleigh quotient optimization problem related to the Maynard-Tao sieve with the goal of finding a more efficient function basis. We demonstrate that a family based on products of reciprocals, previously used in asymptotic estimates for Hm = liminfn → ∞(pn+m−pn), provides results that are close to the theoretical limits of the sieve across many regimes. In particular, we establish H2 ≤ 31 140, H3 ≤ 2 560 022, H4 ≤ 193 672 056, and H5 ≤ 13 659 045 840. We also show that, as k → ∞, the related variational quantity Mk, introduced by Maynard, satisfies the lower bound Mk ≥ log k − 0.273572 + o(1).

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Cite

BibTeXRIS
BibTeX
@misc{lewulis2026clusters,
  title = {{Clusters of primes revisited}},
  author = {Paweł Lewulis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41113},
  eprint = {2610.41113},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41113},
}
RIS
TY  - UNPB
TI  - Clusters of primes revisited
AU  - Paweł Lewulis
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41113
UR  - https://arxiv.org/abs/2610.41113
AB  - We revisit the generalized Rayleigh quotient optimization problem related to the Maynard-Tao sieve with the goal of finding a more efficient function basis. We demonstrate that a family based on products of reciprocals, previously used in asymptotic estimates for H_m = liminf_n → ∞(p_n+m−p_n), provides results that are close to the theoretical limits of the sieve across many regimes. In particular, we establish H_2 ≤ 31 140, H_3 ≤ 2 560 022, H_4 ≤ 193 672 056, and H_5 ≤ 13 659 045 840. We also show that, as k → ∞, the related variational quantity M_k, introduced by Maynard, satisfies the lower bound M_k ≥ log k − 0.273572 + o(1).
ER  -