# Primes in tuples II

@article{Goldston2007PrimesIT,
title={Primes in tuples II},
author={D. A. Goldston and Janos Pintz and Cem Yalçin Yıldırım},
journal={Acta Mathematica},
year={2007},
volume={204},
pages={1-47}
}
• Published 15 October 2007
• Mathematics
• Acta Mathematica
AbstractWe prove that $$\mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty,$$where pn denotes the nth prime. Since on average pn+1−pn is asymptotically log n, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p−p′ between primes which includes the small gap result above.
A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields
D. A. Goldston, J. Pintz and C. Y. Yıldırım  proved the existence of infinitely many prime pairs whose difference is arbitrarily small compared to the average gap, namely $$\liminf_{n\to\infty} Limit Points of the Sequence of Normalized Differences Between Consecutive Prime Numbers • Mathematics • 2015 Let p n denote the nth prime number and let $$d_{n} = p_{n+1} - p_{n}$$ denote the nth difference in the sequence of prime numbers. Erdős and Ricci independently proved that the set of limit points Bounded gaps between primes in short intervals • Mathematics • 2017 Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form$$[x-x^{0.525},x][x-x0.525,x] for large x. In this paper, we extend a result of Maynard
On a conjecture of Erd\H{O}s, P\'olya and Tur\'an on consecutive gaps between primes
Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote the sequence of all gaps between consecutive primes. In 1948 Erd\H{o}s and Tur\'an showed that d_{n+1}-d_n changes sign
On the Ratio of Consecutive Gaps Between Primes
In the present work we prove a common generalization of Maynard–Tao’s recent result about consecutive bounded gaps between primes and of the Erdős–Rankin bound about large gaps between consecutive
Note On The Maximal Primes Gaps
This note presents a result on the maximal prime gap of the form p_(n+1) - p_n 0 is a constant, for any arbitrarily small real number e > 0, and all sufficiently large integer n > n_0. Equivalently,
Fractional Parts Of Non-Integer Powers Of Primes. II
Let $\alpha > 0$ be any fixed non-integer, $I$ be any subinterval of $[0; 1)$. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for the set of primes $p$ satisfying the condition $\{ Prime numbers in logarithmic intervals • Mathematics • 2009 Let X be a large parameter. We will first give a new estimate for the i ntegral moments of primes in short intervals of the type (p, p + h], where p ≤ X is a prime number and h = o(X). Then we will Bounded prime gaps in short intervals We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval$[x,x+x (\log x)^{-A}]\$. Our