# Short intervals with a given number of primes

@article{Freiberg2015ShortIW,
title={Short intervals with a given number of primes},
author={Tristan Freiberg},
journal={arXiv: Number Theory},
year={2015}
}
• T. Freiberg
• Published 1 August 2015
• Mathematics
• arXiv: Number Theory

### A Note on the Distribution of Primes in Intervals

Assuming a certain form of the Hardy–Littlewood prime tuples conjecture, we show that, given any positive numbers λ1, …, λr and nonnegative integers m1, …, mr, the proportion of positive integers \(n

### POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES

We prove that for every $m\geq 0$ there exists an $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if $0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and $x$ is sufficiently large in

### Almost primes in almost all short intervals

• Joni Teräväinen
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2016
Abstract Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵ x] contain E 3 numbers, and almost all intervals [x,x + log3.51 x]

### Almost Primes in Almost All Short Intervals

Let E k be the set of positive integers having exactly k prime factors. We show that almost all intervals [ x, x +log 1+ ε x ] contain E 3 numbers, and almost all intervals [ x, x +log 3 . 51 x ]