## A NOTE ON TWIN PRIMES (2005)

### BibTeX

@MISC{Goldston05anote,

author = {D. A. Goldston},

title = {A NOTE ON TWIN PRIMES},

year = {2005}

}

### OpenURL

### Abstract

ABSTRACT. We relate the twin prime conjecture to corresponding conjectures for a short divisor sum which approximates primes. The twin prime conjecture states that there are infinitely many pairs of primes differing by two. More generally we expect there will be infinitely many pairs of primes with difference k, for any fixed even integer k. Let ƒ.n / be the von Mangoldt function defined to be log p if n D pm, m 1, and zero otherwise. Then a quantitative version of the general twin prime conjecture is that, as N!1, NX (1) ƒ.n/ƒ.n C k / D.S.k / C o.1//N; nD1 where S.k / is the singular series given by 8

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Citation Context ... expressions where ƒR.n/ or a combination of ƒR.n/’s are squared. In [2, 3, 4] C. Y. Yıldırım and I obtained asymptotic formulas using ƒR.n/ to approximate the Hardy-Littlewood prime tuple conjecture =-=[6]-=-. In particular, we proved the following results which are the two simplest approximations of (1) with non-negative weights. Suppose 1 R N 1 4 for any >0 and 1 k R 1 8 , then for R; N !1, .A/ NX ƒ.n/.... |

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Citation Context ... NSF Grant and the American Institute of Mathematics. 1s2 D. A. GOLDSTON retain on average some of the same properties as ƒR.n/. In this round-about way we can obtain new results on primes themselves =-=[2, 4]-=-. These applications require positivity, but ƒR.n/ is not non-negative – actually it is frequently negative, and therefore we use expressions where ƒR.n/ or a combination of ƒR.n/’s are squared. In [2... |

14 |
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Citation Context ... main error term, but there are interesting techniques available for non-trivial estimates. In the first place, Graham [5] proved , for 1 R N , (9) NX .ƒR.n// 2 D N log R C O.N /: nD1 Recently Hooley =-=[7]-=- introduced a new method for a closely related problem which also applies for (9) and should generalize to situations such as .B/. More fundamentally, one must deal non-trivially with k ¤ 0. Here Iwan... |

12 |
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Citation Context ... NSF Grant and the American Institute of Mathematics. 1s2 D. A. GOLDSTON retain on average some of the same properties as ƒR.n/. In this round-about way we can obtain new results on primes themselves =-=[2, 4]-=-. These applications require positivity, but ƒR.n/ is not non-negative – actually it is frequently negative, and therefore we use expressions where ƒR.n/ or a combination of ƒR.n/’s are squared. In [2... |

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7 |
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(Show Context)
Citation Context ...ge 1 R N 1 4 where .B/ holds. This result is obtained by trivially estimating the main error term, but there are interesting techniques available for non-trivial estimates. In the first place, Graham =-=[5]-=- proved , for 1 R N , (9) NX .ƒR.n// 2 D N log R C O.N /: nD1 Recently Hooley [7] introduced a new method for a closely related problem which also applies for (9) and should generalize to situations s... |

1 |
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(Show Context)
Citation Context ...4]. These applications require positivity, but ƒR.n/ is not non-negative – actually it is frequently negative, and therefore we use expressions where ƒR.n/ or a combination of ƒR.n/’s are squared. In =-=[2, 3, 4]-=- C. Y. Yıldırım and I obtained asymptotic formulas using ƒR.n/ to approximate the Hardy-Littlewood prime tuple conjecture [6]. In particular, we proved the following results which are the two simplest... |