## Small gaps between primes

Citations: | 7 - 3 self |

### BibTeX

@MISC{Goldston_smallgaps,

author = {D. A. Goldston and C. Y. Yildirim},

title = {Small gaps between primes},

year = {}

}

### OpenURL

### Abstract

ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.

### Citations

154 |
Sieve methods
- Halberstam, Richert
- 1974
(Show Context)
Citation Context ...d therefore the distribution function for prime gaps must spread out from the average. This method gives the result (1.5) ∆ ≤ 1 − 1 2B . The value B = 4 of Bombieri and Davenport [1] (see also [9] or =-=[11]-=-) or B = 3.5 of Bombieri, Friedlander, and Iwaniec [2] 2 , or even slightly smaller values may be used here. 3. The Maier Method. In 1988 Maier [15] found certain intervals (rather sparsely distribute... |

150 | The primes contain arbitrarily long arithmetic progressions
- Green, Tao
(Show Context)
Citation Context ...of Laguerre polynomials needed in our method. The first-named author also thanks the American Institute of Mathematics where much of the collaboration mentioned above took place. In a recent preprint =-=[10]-=- Ben Green and Terence Tao proved a landmark result on4 D. A. GOLDSTON AND C. Y. YILDIRIM arithmetic progressions of primes. One tool they used was the current Proposition 1 from an earlier (not wide... |

133 |
Some problems on partitio numerorum III : On the expression of a number as a sum of primes
- Littlewood
- 1923
(Show Context)
Citation Context ...is admissible if and only if νp(H) < p for all p. Letting Λ(n) denote the von Mangoldt function, define (2.3) Λ(n; H) = Λ(n + h1)Λ(n + h2) · · · Λ(n + hk). The Hardy-Littlewood prime tuple conjecture =-=[12]-=- states that for H admissible, ∑ (2.4) Λ(n; H) = N ( S(H) + o(1) ) , as N → ∞. n≤N (This is trivially true if H is not admissible.) We approximate Λ(n) as in our earlier work by using the truncated di... |

65 |
The Theory of the Riemann Zeta-function, Second Edition revised by D.R
- Titchmarsh
- 1988
(Show Context)
Citation Context ...rywhere except for a simple pole with residue 1 at s = 1, and therefore (3.2) ζ(s) − 1 s − 1 is an entire function. We need to use a classical zero-free region result. By Theorem 3.11 and (3.12.8) of =-=[20]-=- there exists a small positive constant C such that ζ(σ + it) ̸= 0 in the region (3.3) σ ≥ 1 − for all t, and further (3.4) ζ(σ + it) − C log(|t| + 2) 1 ≪ log(|t| + 2), σ − 1 + it 1 ≪ log(|t| + 2), ζ(... |

35 |
On the difference between consecutive primes
- Huxley
- 1972
(Show Context)
Citation Context ...≤ e −γ = 0.56145 . . . . In contrast to the first two methods, this method does not produce a positive proportion of small prime gaps. These methods may be combined to obtain improved results. Huxley =-=[13, 14]-=- combined the first two methods making use of a weighted version of the first method to find (1.7) ∆ ≤ 0.44254 . . . (using B = 4), ∆ ≤ 0.43494 . . . (using B = 3.5), and Maier combined his method wit... |

31 |
Primes in arithmetic progressions to large moduli
- Bombieri, Friedlander, et al.
- 1989
(Show Context)
Citation Context ...st spread out from the average. This method gives the result (1.5) ∆ ≤ 1 − 1 2B . The value B = 4 of Bombieri and Davenport [1] (see also [9] or [11]) or B = 3.5 of Bombieri, Friedlander, and Iwaniec =-=[2]-=- 2 , or even slightly smaller values may be used here. 3. The Maier Method. In 1988 Maier [15] found certain intervals (rather sparsely distributed) where there are e γ more primes than the expected n... |

28 | Higher correlations of divisor sums related to primes. I. Triple correlations
- Goldston, Yıldırım
(Show Context)
Citation Context ...h improved and made unconditional. This approach can be interpreted as a second moment method using a truncated divisor sum as an approximation of Λ(n), the von Mangoldt function (see introduction in =-=[7]-=-). The method proves (1.3) ∆ ≤ 1 2 . ) , Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary 11N05 ; Secondary 11P32. Key words and phrases. prime number. Goldston was supported b... |

28 |
The difference between consecutive prime numbers
- Rankin
- 1938
(Show Context)
Citation Context ... was supported by NSF; Yıldırım was supported by TÜB˙ITAK. 1 In the unpublished paper Partitio Numerorum VII they proved, assuming the Generalized Riemann Hypothesis, that ∆ ≤ 2 1+4Θ . In 1940 Rankin =-=[18]-=- refined Hardy and Littlewood’s method to show that ∆ ≤ , 3 5 where Θ is the supremum of the real parts of all the zeros of all Dirichlet L-functions. In particular assuming the Generalized Riemann Hy... |

24 |
On the difference of consecutive primes
- Erdos
- 1935
(Show Context)
Citation Context ...s. In particular assuming the Generalized Riemann Hypothesis (Θ = 1 3 ) this gives ∆≤ 2 5 . 12 D. A. GOLDSTON AND C. Y. YILDIRIM 2. The Erdös Method. By the prime number theorem we have ∆ ≤ 1. Erdös =-=[4]-=- in 1940 was the first to prove unconditionally that ∆ < 1. He used the sieve upper bound for primes differing by an even number k ∑ (1.4) Λ(n)Λ(n + k) ≤ (B + ǫ)S(k)N n≤N where S(k) is the singular se... |

22 |
On the distribution of primes in short intervals, Mathematika 23
- Gallagher
- 1976
(Show Context)
Citation Context ...of this length. We therefore let (7.1) ψ(x) = ∑ Λ(n), n≤x (7.2) ψ(n, h) = ψ(n + h) − ψ(n), (7.3) h = λlog N, and in this paper we assume that (7.4) λ ≪ 1. The model for our method is due to Gallagher =-=[5]-=-, who proved that if the Hardy-Littlewood conjecture (2.4) holds uniformly for h ≪ log N then one can asymptotically evaluate all the moments for the number of primes in intervals of length h. Thus as... |

22 |
Theory of Equations, McGraw-Hill
- Uspensky
- 1948
(Show Context)
Citation Context ... · · + ckak−1 = −ck+1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ck−1a0 + cka1 + ck+1a2 + · · · + c2k−2ak−1 = −c2k−1 we have by Cramer’s rule (see =-=[21]-=-) that these equations have the solution (6.20) aj = − D(j+1) k−1 Dk−1 , j = 0, 1, . . ., k − 1, provided that Dk−1 ̸= 0, where D (i) k−1 is the determinant with the ith column of Dk−1 replaced by the... |

21 |
Small difference between prime
- Bombieri, Davenport
- 1966
(Show Context)
Citation Context ...mates for ∆. 1. The Hardy-Littlewood and Bombieri-Davenport Method. In the mid-1920’s Hardy and Littlewood 1 used the circle method to obtain a conditional result which in 1965 Bombieri and Davenport =-=[1]-=- both improved and made unconditional. This approach can be interpreted as a second moment method using a truncated divisor sum as an approximation of Λ(n), the von Mangoldt function (see introduction... |

19 |
differences between prime numbers
- Small
(Show Context)
Citation Context ... = 4 of Bombieri and Davenport [1] (see also [9] or [11]) or B = 3.5 of Bombieri, Friedlander, and Iwaniec [2] 2 , or even slightly smaller values may be used here. 3. The Maier Method. In 1988 Maier =-=[15]-=- found certain intervals (rather sparsely distributed) where there are e γ more primes than the expected number, and therefore within these intervals the average spacing is reduced by a factor of e −γ... |

13 | Some new asymptotic properties for the zeros of Jacobi, Laguerre and Hermite polynomials
- Dette, Studden
- 1995
(Show Context)
Citation Context ...β(n) − n and limn→∞ β(n) n = A > 0, then x1(n, α) (6.36) lim = ( n→∞ n √ A − 1) 2 . Proof. The properties of Ln (α) (x) may be found in Szegö [19]. Equation (6.36) is a special case of Theorem 4.4 of =-=[3]-=-. A simple proof may be obtained by using the same argument found in [16] where a result corresponding to (6.36) for Jacobi polynomials is proved using Sturm comparison theory. By (5.1.2) of Szegö, th... |

5 |
An application of the Fouvry-Iwaniec theorem, Acta Arithmetica XLIII
- Huxley
- 1984
(Show Context)
Citation Context ...≤ e −γ = 0.56145 . . . . In contrast to the first two methods, this method does not produce a positive proportion of small prime gaps. These methods may be combined to obtain improved results. Huxley =-=[13, 14]-=- combined the first two methods making use of a weighted version of the first method to find (1.7) ∆ ≤ 0.44254 . . . (using B = 4), ∆ ≤ 0.43494 . . . (using B = 3.5), and Maier combined his method wit... |

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

2 |
On the zeros of Jacobi polynomials Pn (αn,βn
- Moak, Saff, et al.
- 1979
(Show Context)
Citation Context ... A − 1) 2 . Proof. The properties of Ln (α) (x) may be found in Szegö [19]. Equation (6.36) is a special case of Theorem 4.4 of [3]. A simple proof may be obtained by using the same argument found in =-=[16]-=- where a result corresponding to (6.36) for Jacobi polynomials is proved using Sturm comparison theory. By (5.1.2) of Szegö, the differential equation (6.37) u ′′ ( n + (α + 1)/2 + + x has u = e −x/2 ... |

2 |
Beyond pair correlation to appear
- Montgomery, Soundararajan
(Show Context)
Citation Context ...e fix k ≥ 1; the argument works equally well for any k, and we can take k = 1 if we wish. Suppose now that R = N 1 4k (log N) −B(k) , h ≪ log N. 3 Granville (unpublished) and Montgomery-Soundararajan =-=[17]-=- have recently proved more precise results, but these are not needed here.6 D. A. GOLDSTON AND C. Y. YILDIRIM By differencing, equation (2.11) continues to hold when the sum on the left-hand side is ... |

1 |
Sieves in Number Theory Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete Vol
- Greaves
- 2001
(Show Context)
Citation Context ...nce, and therefore the distribution function for prime gaps must spread out from the average. This method gives the result (1.5) ∆ ≤ 1 − 1 2B . The value B = 4 of Bombieri and Davenport [1] (see also =-=[9]-=- or [11]) or B = 3.5 of Bombieri, Friedlander, and Iwaniec [2] 2 , or even slightly smaller values may be used here. 3. The Maier Method. In 1988 Maier [15] found certain intervals (rather sparsely di... |