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
View PDF on arXiv
8 Citations
Highly Influential Citations
1
Background Citations
5
Results Citations
1

8 Citations

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…

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

TLDR
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…

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 ]…

References

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Dense clusters of primes in subsets

  • J. Maynard
  • Mathematics, Computer Science
    Compositio Mathematica
  • 2016
TLDR
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

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

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…

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

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…

Small gaps between primes

The twin prime conjecture states that there are infinitely many pairs of distinct primes which differ by 2. Until recently this conjecture had seemed to be out of reach with current techniques.…

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…

Equidistribution in number theory, an introduction

Preface. Contributors.- Biographical Sketches of the Lecturers. Uniform Distribution A. Granville and Z. Rudnick.- 1. Uniform Distribution mod One.2. Fractional Parts of an2 .3. Uniform Distribution…

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…