Almost primes in almost all very short intervals

: Matomäki Kaisa

Publisher: WILEY

: 2022

: Journal of the London Mathematical Society

: JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES

: J LOND MATH SOC

: 37

: 0024-6107

: 1469-7750

DOI: https://doi.org/10.1112/jlms.12592

: https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12592

: https://research.utu.fi/converis/portal/detail/Publication/175009419

We show that as soon as h ->infinity$h\rightarrow \infty$ with X ->infinity$X \rightarrow \infty$, almost all intervals (x-hlogX,x]$(x-h\log X, x]$ with x is an element of(X/2,X]$x \in (X/2, X]$ 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.

Journal of London Math Soc - 2022 - Matom ki - Almost primes in almost all very short intervals.pdf