## Primes in Tuples I

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Citations: | 6 - 1 self |

### BibTeX

@MISC{Goldston_primesin,

author = {D. A. Goldston and J. Pintz and C. Y. Yıldırım},

title = { Primes in Tuples I},

year = {}

}

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### Abstract

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 Elliott-Halberstam 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).

### Citations

173 |
The Theory of the Riemann Zeta–Function
- Titchmarsh
- 1986
(Show Context)
Citation Context ...C O.js 1j/: s 1 (Here is Euler’s constant.) We need standard information concerning the classical zero-free region of the Riemann zeta-function. By Theorem 3.11 and (3.11.8) inPRIMES IN TUPLES I 835 =-=[32]-=-, there exists a small constant c > 0, for which we assume c 10 2 , such that . C it/ ¤ 0 in the region (5.3) 1 for all t. Furthermore, we have (5.4) . C it/ 1 1 C it 0 . C it/ C 4c log.jtj C 3/ log.j... |

154 |
Sieve methods
- Halberstam, Richert
- 1974
(Show Context)
Citation Context ... (9.9) ν ∗ p(G) = νp(G 0 ) − 1, where (9.10) G 0 = G ∪ {h0}. We extend this definition to ν∗ d (H1 0 ) for squarefree numbers d by multiplicativity. The function ν∗ d is familiar in sieve theory, see =-=[15]-=-. A more algebraic discussion of νd ∗ may also be found in [13, 14]. We define ν∗ ( d (H1∩H2) 0) as in (7.5). Next, the divisibility conditions a2|PH2(n), a12|PH1(n), and a12|PH2(n) are handled as in ... |

133 |
Some problems on partitio numerorum III : On the expression of a number as a sum of primes
- Littlewood
- 1923
(Show Context)
Citation Context ... h2/ ƒ.n C h k/ and use this function to detect prime tuples and tuples with prime powers in components, the latter of which can be removed in applications. The HardyLittlewood prime-tuple conjecture =-=[17]-=- can be stated in the form X (2.4) ƒ.nI �/ D N.S.�/ C o.1//; as N ! 1. n N (This conjecture is trivially true if � is not admissible.) Except for the prime number theorem (1-tuples), this conjecture i... |

116 | Multiplicative number theory - Davenport - 1967 |

82 | Topics in multiplicative number theory - Montgomery - 1971 |

65 |
The Theory of the Riemann Zeta-function, Second Edition revised by D.R
- Titchmarsh
- 1988
(Show Context)
Citation Context ....2) ζ(s) = 1 + γ + O(|s − 1|). s − 1 (Here γ is Euler’s constant.) We need standard information concerning the classical zero-free region of the Riemann zeta-function. By Theorem 3.11 and (3.11.8) in =-=[30]-=-, there exists a small constant c > 0, for which we assume c ≤ 10 −2 , such that ζ(σ + it) ̸= 0 in the region (5.3) σ ≥ 1 − for all t. Furthermore, we have (5.4) ζ(σ + it) − 4c log(|t| + 3) 1 ≪ log(|t... |

58 |
A heuristic asymptotic formula concerning the distribution of prime numbers
- Bateman, Horn
- 1962
(Show Context)
Citation Context ...s we will see in Section 5, this approximation suggests the Hardy-Littlewood type conjecture X (2.11) ƒk.nI �/ D N .S.�/ C o.1// : n N This is a special case of the general conjecture of Bateman–Horn =-=[1]-=- which is the quantitative form of Schinzel’s conjecture [28]. In analogy with (2.6) (when k D 1), we approximate ƒ k.n/ by the smoothed and truncated divisor sum and define (2.12) X djn d R ƒR.nI �/ ... |

55 |
Sur certaines hypothèses concernant les nombres premiers
- Schinzel, Sierpiński
- 1958
(Show Context)
Citation Context ...Hardy-Littlewood type conjecture X (2.11) ƒk.nI �/ D N .S.�/ C o.1// : n N This is a special case of the general conjecture of Bateman–Horn [1] which is the quantitative form of Schinzel’s conjecture =-=[28]-=-. In analogy with (2.6) (when k D 1), we approximate ƒ k.n/ by the smoothed and truncated divisor sum and define (2.12) X djn d R ƒR.nI �/ D 1 kŠ .d/ X djP�.n/ d R log R d .d/ k k ; log R d However, a... |

35 |
On the difference between consecutive primes
- Huxley
- 1972
(Show Context)
Citation Context ... with the method of Erdős and obtained E1 ≤ 0.4665 . . .. Their result was further refined by Pilt’ai [24] to E1 ≤ 0.4571 . . ., Uchiyama [31] to E1 ≤ 0.4542 . . . and in several steps by Huxley [19] =-=[20]-=- to yield E1 ≤ 0.4425 . . ., and finally in 1984 to E1 ≤ .4393 . . . [21]. In 1988 Maier [22] used his matrix-method to improve Huxley’s result to E1 ≤ e −γ · 0.4425 · · · = 0.2484 . . ., where γ is E... |

31 |
Primes in arithmetic progressions to large moduli
- Bombieri, Friedlander, et al.
- 1989
(Show Context)
Citation Context ..., any improvement in the level of distribution # beyond 1=2 probably lies very deep, and even the GRH does not help. Still, there are stronger versions of the Bombieri-Vinogradov theorem, as found in =-=[3]-=-, and the circle of ideas used to prove these results, which may help to obtain this result. Question 2. Is # D 1=2 a true barrier for obtaining primes in tuples? Soundararajan [31] has demonstrated t... |

30 | Multiplicative Number Theory, Second Edition, revised by H.L - Davenport - 1980 |

28 | Higher correlations of divisor sums related to primes. I. Triple correlations
- Goldston, Yıldırım
(Show Context)
Citation Context ...ey’s result to E1 ≤ e −γ · 0.4425 · · · = 0.2484 . . ., where γ is Euler’s constant. Maier’s method by itself gives E1 ≤ e −γ = 0.5614 . . .. The recent version of the method of Goldston and Yıldırım =-=[12]-=- led, without combination with other methods, to E1 ≤ 1/4. In a later paper in this series we will prove the quantitative result that (1.9) liminf n→∞ pn+1 − pn (log pn) 1 < ∞. 2 (log log pn) 2 While ... |

28 |
The difference between consecutive prime numbers
- Rankin
- 1938
(Show Context)
Citation Context ... theorem. The first result of type �1 < 1 was proved in 1926 by Hardy and Littlewood [18], who on assuming the Generalized Riemann Hypothesis (GRH) obtained �1 2=3. This result was improved by Rankin =-=[26]-=- to �1 3=5; also assuming the GRH. The first unconditional estimate was proved by Erdős [7] in 1940. Using Brun’s sieve, he showed that �1 < 1 c with an unspecified positive explicitly calculable cons... |

24 |
On the difference of consecutive primes
- Erdos
- 1935
(Show Context)
Citation Context ...ho on assuming the Generalized Riemann Hypothesis (GRH) obtained �1 2=3. This result was improved by Rankin [26] to �1 3=5; also assuming the GRH. The first unconditional estimate was proved by Erdős =-=[7]-=- in 1940. Using Brun’s sieve, he showed that �1 < 1 c with an unspecified positive explicitly calculable constant c. His estimate was improved by Ricci [27] in 1954 to �1 15=16: In 1965, Bombieri and ... |

22 |
On the distribution of primes in short intervals, Mathematika 23
- Gallagher
- 1976
(Show Context)
Citation Context ...R(n; Hk, ℓ) 2 , where r is a positive integer. To evaluate ˜ S, we need the case of Proposition 2 where h0 ̸∈ Hk, (3.6) ∑ ΛR(n; Hk, ℓ) 2 θ(n + h0) ∼ n≤N 1 (k + 2ℓ)! We also need a result of Gallagher =-=[9]-=-: as h → ∞, ( ) 2ℓ S(Hk ∪ {h0})N(log R) ℓ k+2ℓ . (3.7) ∑ 1≤h1,h2,...hk≤h distinct S(Hk) ∼ h k . Taking R = N ϑ 2 −ε , and applying (3.1), (3.2), (3.6), and (3.7), we find that (3.8) ˜S ∼ ∑ 1≤h1,h2,...... |

21 |
Small difference between prime
- Bombieri, Davenport
- 1966
(Show Context)
Citation Context ...ng Brun’s sieve, he showed that E1 < 1 − c with an unspecified positive explicitly calculable constant c. His estimate was improved by Ricci [26] in 1954 to E1 ≤ 15/16. In 1965 Bombieri and Davenport =-=[2]-=- refined and made unconditional the method of Hardy and Littlewood by substituting the Bombieri–Vinogradov theorem for the GRH, and obtained E1 ≤ 1/2. They also combined their method with the method o... |

19 |
differences between prime numbers
- Small
(Show Context)
Citation Context ... : and in several steps by Huxley [20], [21] to yield �1 0:4425 : : : , and finally in 1984 to �1 :4393 : : : [22]. This was further improved by Fouvry and Grupp [9] to �1 :4342 : : : : In 1988 Maier =-=[23]-=- used his matrix-method to improve Huxley’s result to �1 e 0:4425 : : : D 0:2484 : : : , where is Euler’s constant. Maier’s method by itself gives �1 e D 0:5614 : : : . The recent version of the metho... |

16 |
A conjecture in prime number theory
- Elliott, Halberstam
(Show Context)
Citation Context ...1.3) max N a ˇ .N I q; a/ N .q/ ˇ : .log N / A q Q .a;q/D1 We say that the primes have level of distribution # if (1.3) holds for any A > 0 and any " > 0 with (1.4) Q D N # " : Elliott and Halberstam =-=[5]-=- conjectured that the primes have level of distribution 1. According to the Bombieri-Vinogradov theorem, the primes are known to have level of distribution 1=2. Let n be a natural number and consider ... |

14 |
On the distribution of primes in short intervals
- Gallagher
- 1976
(Show Context)
Citation Context ...f Proposition 2 where h0 62 �k: (3.6) X ƒR.nI �k; `/ 2 .nCh0/ n N 1 .kC2`/Š 2` ` S.� k[fh0g/N.log R/ kC2` :830 DANIEL A. GOLDSTON, JÁNOS PINTZ, and CEM Y. YILDIRIM We also need a result of Gallagher =-=[10]-=-: as h ! 1, X (3.7) S.�k/ h k : 1 h1;h2;:::hk h distinct Taking R D N # 2 " , and applying (3.1), (3.2), (3.6), and (3.7), we find that (3.8) z� X 1 h1;h2;:::;hk h distinct k .k C 2` C 1/Š C X 1 h0 h ... |

11 | gaps between prime numbers: the work of Goldston-Pintz-Yıldırım
- Small
(Show Context)
Citation Context ...heorem, as found in [3], and the circle of ideas used to prove these results, which may help to obtain this result. Question 2. Is # D 1=2 a true barrier for obtaining primes in tuples? Soundararajan =-=[31]-=- has demonstrated this is the case for the current argument, but perhaps more efficient arguments may be devised. Question 3. Assuming the Elliott-Halberstam conjecture, can it be proved that there ar... |

9 |
Small differences between prime numbers
- Bombieri, Davenport
- 1966
(Show Context)
Citation Context ...Using Brun’s sieve, he showed that �1 < 1 c with an unspecified positive explicitly calculable constant c. His estimate was improved by Ricci [27] in 1954 to �1 15=16: In 1965, Bombieri and Davenport =-=[2]-=- refined and made unconditional the method of Hardy and Littlewood by substituting the Bombieri-Vinogradov theorem for the GRH, and obtained �1 1=2. They also combined their method with the method of ... |

8 | Small gaps between primes or almost primes
- Goldston, Graham, et al.
(Show Context)
Citation Context ...subject, we have two other papers that overlap some of the results here. The first paper [15], written jointly with Motohashi, gives a short and simplified proof of Theorems 1 and 2. The second paper =-=[14]-=-, written jointly with Graham, uses sieve methods to prove Theorems 1 and 2 and provides applications for tuples of almost-primes (products of a bounded number of primes.) The present paper is organiz... |

7 | Yildirim, Small gaps between primes exist
- Goldston, Motohashi, et al.
(Show Context)
Citation Context ...roofs of these results will appear in later papers in this series. While this paper is our first paper on this subject, we have two other papers that overlap some of the results here. The first paper =-=[15]-=-, written jointly with Motohashi, gives a short and simplified proof of Theorems 1 and 2. The second paper [14], written jointly with Graham, uses sieve methods to prove Theorems 1 and 2 and provides ... |

6 |
Zero-free regions for the Riemann zeta function, Number Theory for the Millennium
- Ford
- 2002
(Show Context)
Citation Context ...C it 0 . C it/ C 4c log.jtj C 3/ log.jtj C 3/; 1 1 C it 1 . C it/ log.jtj C 3/; log.jtj C 3/; in this region. We will fix this c for the rest of the paper (we could take, for instance, c D 10 2 , see =-=[8]-=-). Let � denote the contour given by (5.5) s D c C it: log.jtj C 3/ LEMMA 1. For R C , k 2, B C k, Z (5.6) .log.jsj C 3// B R ˇ s ds sk ˇ C k 1 R c2 p c log C e R=2; � where C1; c2 and the implied con... |

6 |
On the switching principle in sieve theory
- Fouvry, Grupp
- 1986
(Show Context)
Citation Context ...: : : , Uchiyama [33] to �1 0:4542 : : : and in several steps by Huxley [20], [21] to yield �1 0:4425 : : : , and finally in 1984 to �1 :4393 : : : [22]. This was further improved by Fouvry and Grupp =-=[9]-=- to �1 :4342 : : : : In 1988 Maier [23] used his matrix-method to improve Huxley’s result to �1 e 0:4425 : : : D 0:2484 : : : , where is Euler’s constant. Maier’s method by itself gives �1 e D 0:5614 ... |

6 |
gaps between primes or almost primes, preprint available at http://www.arxiv.org/abs/math/0506067
- Goldston, Graham, et al.
(Show Context)
Citation Context ...first paper on this subject, we have two other papers which overlaps it. The first paper [14], written jointly with Motohashi, gives a short and simplified proof of Theorems 1 and 2. The second paper =-=[13]-=-, written jointly with Graham, uses sieve methods to prove Theorems 1 and 2 and provides applications for tuples of almost-primes (products of two or more distinct primes.) The present paper is organi... |

5 |
Almost-prime k-tuples, Mathematika 44
- Heath-Brown
- 1997
(Show Context)
Citation Context ...d (2.7) to detect small gaps between primes and proved pnC1 pn 1 �1 D lim inf n!1 log pn 4 : In this paper we introduce a new approximation, the idea for which came partly from a paper of Heath-Brown =-=[19]-=- on almost prime tuples. His result is itself a generalization of Selberg’s proof from 1951 (see [29, pp. 233–245]) that the polynomial n.n C 2/ will infinitely often have at most five distinct prime ... |

5 |
On the differences of primes in arithmetical progressions
- Huxley
(Show Context)
Citation Context ...ir method with the method of Erdős and obtained �1 0:4665 : : : . Their result was further refined by Pilt’ai [25] to �1 0:4571 : : : , Uchiyama [33] to �1 0:4542 : : : and in several steps by Huxley =-=[20]-=-, [21] to yield �1 0:4425 : : : , and finally in 1984 to �1 :4393 : : : [22]. This was further improved by Fouvry and Grupp [9] to �1 :4342 : : : : In 1988 Maier [23] used his matrix-method to improve... |

5 |
An application of the Fouvry-Iwaniec theorem, Acta Arithmetica XLIII
- Huxley
- 1984
(Show Context)
Citation Context ...s further refined by Pilt’ai [24] to E1 ≤ 0.4571 . . ., Uchiyama [31] to E1 ≤ 0.4542 . . . and in several steps by Huxley [19] [20] to yield E1 ≤ 0.4425 . . ., and finally in 1984 to E1 ≤ .4393 . . . =-=[21]-=-. In 1988 Maier [22] used his matrix-method to improve Huxley’s result to E1 ≤ e −γ · 0.4425 · · · = 0.2484 . . ., where γ is Euler’s constant. Maier’s method by itself gives E1 ≤ e −γ = 0.5614 . . ..... |

5 |
Sull’andamento della differenza di numeri primi consecutivi
- Ricci
- 1954
(Show Context)
Citation Context ...t unconditional estimate was proved by Erdős [7] in 1940. Using Brun’s sieve, he showed that �1 < 1 c with an unspecified positive explicitly calculable constant c. His estimate was improved by Ricci =-=[27]-=- in 1954 to �1 15=16: In 1965, Bombieri and Davenport [2] refined and made unconditional the method of Hardy and Littlewood by substituting the Bombieri-Vinogradov theorem for the GRH, and obtained �1... |

4 | On Bombieri and Davenport’s theorem concerning small gaps between primes, Mathematika 39 - Goldston - 1992 |

4 |
Méthodes de crible appliquées aux sommes de Kloosterman et aux petits écarts entre nombres premiers, Thése de Doctorat de l’Université
- Sivak
- 2005
(Show Context)
Citation Context ...hat s1 C s2 be restricted to the region to the right of � if we wish to use the bounds in (5.4). 7 By 7 This was pointed out to us by J. Sivak and also Y. Motohashi and was handled in similar ways in =-=[30]-=- and in [15]; we have also adopted this approach here.PRIMES IN TUPLES I 845 the conditions of Lemma 3, (5.4), and (8.3), we have (8.7) M D.s1; s2/ s uC1 1 s vC1 2 .s1Cs2/ d exp.CM U ı1Cı2 log log U ... |

3 |
On the size of the difference between consecutive primes, Issledovania po teorii chisel
- Pilt’ai
- 1972
(Show Context)
Citation Context ...g the Bombieri-Vinogradov theorem for the GRH, and obtained �1 1=2. They also combined their method with the method of Erdős and obtained �1 0:4665 : : : . Their result was further refined by Pilt’ai =-=[25]-=- to �1 0:4571 : : : , Uchiyama [33] to �1 0:4542 : : : and in several steps by Huxley [20], [21] to yield �1 0:4425 : : : , and finally in 1984 to �1 :4393 : : : [22]. This was further improved by Fou... |

3 |
On the difference between consecutive prime numbers
- Uchiyama
- 1975
(Show Context)
Citation Context ...or the GRH, and obtained �1 1=2. They also combined their method with the method of Erdős and obtained �1 0:4665 : : : . Their result was further refined by Pilt’ai [25] to �1 0:4571 : : : , Uchiyama =-=[33]-=- to �1 0:4542 : : : and in several steps by Huxley [20], [21] to yield �1 0:4425 : : : , and finally in 1984 to �1 :4393 : : : [22]. This was further improved by Fouvry and Grupp [9] to �1 :4342 : : :... |

3 |
An application of the Fouvry-Iwaniec theorem
- HUXLEY
- 1984
(Show Context)
Citation Context ...lt was further refined by Pilt’ai [25] to �1 0:4571 : : : , Uchiyama [33] to �1 0:4542 : : : and in several steps by Huxley [20], [21] to yield �1 0:4425 : : : , and finally in 1984 to �1 :4393 : : : =-=[22]-=-. This was further improved by Fouvry and Grupp [9] to �1 :4342 : : : : In 1988 Maier [23] used his matrix-method to improve Huxley’s result to �1 e 0:4425 : : : D 0:2484 : : : , where is Euler’s cons... |

2 |
correlations of divisor sums related to primes, III: Small gaps between primes
- Higher
(Show Context)
Citation Context ... : , where is Euler’s constant. Maier’s method by itself gives �1 e D 0:5614 : : : . The recent version of the method822 DANIEL A. GOLDSTON, JÁNOS PINTZ, and CEM Y. YILDIRIM of Goldston and Yıldırım =-=[13]-=- led, without combination with other methods, to �1 1=4. In a later paper in this series we will prove the quantitative result that (1.9) lim inf n!1 pnC1 pn .log pn/ 1 2 .log log pn/ 2 < 1: While The... |

2 |
differences between consecutive primes
- Small
- 1977
(Show Context)
Citation Context ...hod with the method of Erdős and obtained �1 0:4665 : : : . Their result was further refined by Pilt’ai [25] to �1 0:4571 : : : , Uchiyama [33] to �1 0:4542 : : : and in several steps by Huxley [20], =-=[21]-=- to yield �1 0:4425 : : : , and finally in 1984 to �1 :4393 : : : [22]. This was further improved by Fouvry and Grupp [9] to �1 :4342 : : : : In 1988 Maier [23] used his matrix-method to improve Huxle... |

1 |
A heuristic formula concerning the distribution of prime numbers
- Bateman, Horn
- 1962
(Show Context)
Citation Context ...As we will see in Section 5, this approximation suggests the Hardy–Littlewood type conjecture ∑ (2.11) Λk(n; H) = N (S(H) + o(1)). n≤N This is a special case of the general conjecture of Bateman–Horn =-=[1]-=- which is the quantitative form of Schinzel’s conjecture [27]. There is not much difference between (2.4) and (2.11), but the same is not true of their approximations. In analogy with (2.6) (when k = ... |

1 |
k-tuple permissible patterns
- Engelsma
- 2005
(Show Context)
Citation Context ...m (3.4). For a certain #, it gives the smallest k and corresponding smallest ` for which (3.4) is true. Here h.k/ is the shortest length of any admissible k-tuple, which has been computed by Engelsma =-=[6]-=- by exhaustive search for 1 k 305 and covers every value in this table and the next except h.421/, where we have taken the upper bound value from [6]. # k ` h.k/ 1 7 1 20 0.95 8 1 26 0.90 9 1 30 0.85 ... |

1 |
gaps between primes exist, preprint
- Goldston, Motohashi, et al.
(Show Context)
Citation Context ...t (1.12) can be replaced by the stronger inequality (1.15) Er ≤ e −γ ( √ r − √ 2ϑ) 2 . While this paper is our first paper on this subject, we have two other papers which overlaps it. The first paper =-=[14]-=-, written jointly with Motohashi, gives a short and simplified proof of Theorems 1 and 2. The second paper [13], written jointly with Graham, uses sieve methods to prove Theorems 1 and 2 and provides ... |