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} }
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.
66 Citations
A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields
- Mathematics
- 2013
D. A. Goldston, J. Pintz and C. Y. Yıldırım [3] 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
- Mathematics
- 2015
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
- Mathematics
- 2015
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
- Mathematics
- 2013
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
- Mathematics
- 2020
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 $\{…
Distance between natural numbers based on their prime signature
- MathematicsJournal of Number Theory
- 2021
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
- Mathematics
- 2013
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…
References
SHOWING 1-10 OF 48 REFERENCES
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…
SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES
- Mathematics
- 2005
Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some…
The difference of consecutive primes
- Mathematics
- 1940
is greater than (c2/2)n. A simple calculation now shows that the primes satisfying (4) also satisfy the first inequality of (3) i΀ = e(ci) is chosen small enough. The second inequality of (3) is…
Small gaps between primes exist
- Mathematics
- 2005
In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially…
On the difference between consecutive prime numbers
- Mathematics
- 1975
/ = lim inf ̂ 11-tl . »->» log pn The purpose of this paper is to combine the methods used in two earlier papers1 in order to prove the following theorem. Theorem. (1) / = c(l + 40)/5, where c<…
A heuristic asymptotic formula concerning the distribution of prime numbers
- Mathematics
- 1962
Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these…
Very Large Gaps between Consecutive Primes
- Mathematics
- 1997
Abstract LetG(X) denote the largest gap between consecutive primes belowX. Improving earlier results of Erdős, Rankin, Schonhage, and Maier-Pomerance, we prove G(X)⩾(2e γ +o(1)) log Xlog 2 Xlog 4…
Topics in Multiplicative Number Theory
- Mathematics
- 1971
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 prime numbers: The work of Goldston-Pintz-Yildirim
- Mathematics
- 2006
In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are…
On the switching principle in sieve theory.
- Mathematics
- 1986
to study |(ΛΤ-α)η JS?|. This idea, although quite simple, increases the power of sieve theory by transporting a sieve problem to one which may be more practical. This principle (in a more elaborate…