# 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
Short intervals containing a prescribed number of primes
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
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 Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x+ log x] contain E3 numbers, and almost all intervals [x, x + log x] contain E2 numbers.
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
Weighted Average Number of Prime $m$-tuples lying on an Admissible $k$-tuple of Linear Forms
An upper bound is found for the sum of $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})w_{n}), which will depend on an integral of a smooth function and on the singular series of$\mathcal{H}$, which naturally arises in this context. The twin prime conjecture • J. Maynard • Mathematics Japanese Journal of Mathematics • 2019 The Twin Prime Conjecture asserts that there should be infinitely many pairs of primes which differ by 2. Unfortunately this long-standing conjecture remains open, but recently there has been several GAPS BETWEEN PRIMES • J. Maynard • Mathematics Proceedings of the International Congress of Mathematicians (ICM 2018) • 2019 We discuss recent advances on weak forms of the Prime$k\$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps

## References

SHOWING 1-10 OF 10 REFERENCES
Dense clusters of primes in subsets
• J. Maynard
• Mathematics, Computer Science
Compositio Mathematica
• 2016
It is proved that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primesWhich are close together, and bounds hold with some uniformity in the parameters.
Primes in tuples I
• Mathematics
• 2009
We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the
ON THE DISTRIBUTION OF PRIMES IN SHORT INTERVALS
• Mathematics
• 2010
One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + ft], averaged over n ^ JV, tends to the limit A, when JV and h tend to infinity
Small gaps between primes
We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each
The Distribution of Prime Numbers
THIS interesting “Cambridge Tract” is concerned mainly with the behaviour, for large values of x, of the function n(x), which denotes the number of primes not exceeding x. The first chapter gives
On limit points of the sequence of normalized prime gaps
• Mathematics
• 2016
Let pn denote the n th smallest prime number, and let L denote the set of limit points of the sequence {(pn+1−pn)/logpn}n=1∞ of normalized differences between consecutive primes. We show that, for
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value
Multiplicative Number Theory
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The
Goldbach versus de Polignac numbers
In this article we prove a second moment estimate for the Maynard-Tao sieve and give an application to Goldbach and de Polignac numbers. We show that at least one of two nice properties holds. Either
Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind.
• Commentat. Phys.-Math. 5(25):1–37,
• 1931