Improving on the Brun-Titchmarsh Theorem

Olvier Ramaré; Jan Christoph Schlage-Puchta

doi:10.4064/aa131-4-4

Improving on the Brun-Titchmarsh Theorem

Number Theory 2026-07-05 v1

Authors: Olvier Ramaré , Jan Christoph Schlage-Puchta

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Abstract

We prove that the number of primes in an interval of length $N$ is at most $2N/(\log N+3.53)$ when $N$ is large enough. This is obtained through a sieving process which can be seen as a hybrid between the large sieve and the Selberg sieve, and draws on what we call "local models". This paper has been published in {\sc Acta Arithmetica, 2008, vol. 131, nb. 4, pages 351--366}.

Cite

@article{arxiv.2607.04283,
  title  = {Improving on the Brun-Titchmarsh Theorem},
  author = {Olvier Ramaré and Jan Christoph Schlage-Puchta},
  journal= {arXiv preprint arXiv:2607.04283},
  year   = {2026}
}