Results 1  10
of
13
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 ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. E ..."
Abstract

Cited by 36 (4 self)
 Add to MetaCart
(Show Context)
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 ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, pn+1 − pn lim inf =0. n→ ∞ log pn We will quantify this result further in a later paper (see (1.9) below).
Small gaps between prime numbers: the work of GoldstonPintzYıldırım
, 2000
"... ..."
(Show Context)
Variants of the Selberg sieve, and bounded intervals containing many primes
, 2014
"... For any m ≥ 1, let Hm denote the quantity lim infn→∞(pn+m − pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H1 ≤ 4680, and then by Maynard to H1 ≤ 600, who also established for ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
For any m ≥ 1, let Hm denote the quantity lim infn→∞(pn+m − pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H1 ≤ 4680, and then by Maynard to H1 ≤ 600, who also established for the first time a finiteness result for Hm form ≥ 2, and specifically that Hm m3e4m. If one also assumes the ElliottHalberstam conjecture, Maynard obtained the bound H1 ≤ 12, improving upon the previous bound H1 ≤ 16 of Goldston, Pintz, and Yıldırım, as well as the bound Hm m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1 ≤ 246 unconditionally and H1 ≤ 6 under the assumption of the generalized ElliottHalberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1, h2, h3), there are infinitely many n for which at least two of n + h1, n + h2, n + h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem ’ argument of Selberg to show that the H1 ≤ 6 bound is the best possible that one can obtain from purely sievetheoretic considerations. For largerm, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound Hm me
PRIMES IN INTERVALS OF BOUNDED LENGTH
"... Abstract. In April 2013, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, makes the twin prime conjecture look highly plausible (which can be reinterpreted as the conjec ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. In April 2013, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, makes the twin prime conjecture look highly plausible (which can be reinterpreted as the conjecture that one can take B 2) and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable. The original purpose of this talk was to discuss Zhang’s extraordinary work, putting it in its context in analytic number theory, and to sketch a proof of his theorem. Zhang had even proved the result with B 70 000 000. Moreover, a cooperative team, polymath8, collaborating only online, had been able to lower the value of B to 4680. Not only had they been more careful in several difficult arguments in Zhang’s original paper, they had also developed Zhang’s techniques to be both more powerful and to allow a much simpler proof. Indeed the proof of Zhang’s Theorem, that will be given in the writeup of this talk, is based on these developments. In November, inspired by Zhang’s extraordinary breakthrough, James Maynard dra
BOUNDED GAPS BETWEEN PRODUCTS OF PRIMES WITH APPLICATIONS TO IDEAL CLASS GROUPS AND ELLIPTIC CURVES
"... and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their prime factors in a set of primes P. Our result holds for any P of positive density that satis es a SiegelWal sz condition regarding distribution in arithmetic progressions. We also prove a stronger result in the case that P satis es a BombieriVinogradov condition. We were motivated to prove these generalizations because of recent results of Ono [22] and Soundararajan [25]. These generalizations yield applications to divisibility of class numbers, nonvanishing of critical values of Lfunctions, and triviality of ranks of elliptic curves. 1.
Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers
, 2008
"... ..."
(Show Context)
, (2)
"... Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture. ..."
Abstract
 Add to MetaCart
Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture.
J ( ω) = Π ( P−1 − χ ( P)), (2) 2
"... Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture. ..."
Abstract
 Add to MetaCart
Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture.
BOUNDED GAPS BETWEEN PRIMES
"... Abstract. Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had previously seemed intractable. In this article we will discuss Zhang’s extraordinary work, putting it in its context in analytic number theory, and sketch a proof of his theorem. Zhang even proved the result with B 70 000 000. A cooperative team, polymath8, collaborating only online, has been able to lower the value of B to 4680, and it seems plausible that these techniques can be pushed somewhat further, though the limit of these methods seem, for now, to be B 12. Contents
� � � P, � ( P)
"... Using Jiang function we prove Jiang prime ktuple theorem.We find true singular series. Using the examples we prove the HardyLittlewood prime ktuple conjecture with wrong singular series.. Jiang prime ktuple theorem will replace the HardyLittlewood prime ktuple conjecture. (A) Jiang prime ktup ..."
Abstract
 Add to MetaCart
Using Jiang function we prove Jiang prime ktuple theorem.We find true singular series. Using the examples we prove the HardyLittlewood prime ktuple conjecture with wrong singular series.. Jiang prime ktuple theorem will replace the HardyLittlewood prime ktuple conjecture. (A) Jiang prime ktuple theorem with true singular series[1, 2]. We define the prime ktuple equation p, p � ni, (1) where 2 n, i �1, � k�1. i we have Jiang function [1, 2]