English

The Cram\'er conjecture holds with a positive probability

Number Theory 2018-02-26 v1

Abstract

We prove that a positive proportion of the intervals of any fixed scalar multiple of log(X)\log(X) in the dyadic interval [X,2X][X,2X] contain a prime number. We also show that a positive proportion of the congruence classes modulo qq contain a prime number smaller than any fixed scalar multiple of φ(q)log(q).\varphi(q)\log(q).

Keywords

Cite

@article{arxiv.1802.08524,
  title  = {The Cram\'er conjecture holds with a positive probability},
  author = {Naser T. Sardari},
  journal= {arXiv preprint arXiv:1802.08524},
  year   = {2018}
}
R2 v1 2026-06-23T00:31:22.759Z