Almost primes in almost all very short intervals

Kaisa Matomäki

Almost primes in almost all very short intervals

Number Theory 2022-01-19 v3

Authors: Kaisa Matomäki

Abstract

We show that as soon as $h\to \infty$ with $X \to \infty$ , almost all intervals $(x-h\log X, x]$ with $x \in (X/2, X]$ contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.

Keywords

prime numbers

Cite

@article{arxiv.2012.11565,
  title  = {Almost primes in almost all very short intervals},
  author = {Kaisa Matomäki},
  journal= {arXiv preprint arXiv:2012.11565},
  year   = {2022}
}

Comments

Accepted version (to appear in J. Lond. Math. Soc.). Compared to v2 the exposition has been improved following rereree's suggestions

Almost primes in almost all very short intervals

Abstract

Keywords

Cite

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