The Density Hypothesis for Dirichlet <i>L</i>-Series

Vinogradov, Andrey I.

doi:10.1142/9789814542487_0017

Cited by 13 publications

(15 citation statements)

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“…Two further propositions, Theorem 19 and Theorem 20, use the prime distribution results from the 'Distribution estimates on arithmetic functions' section to give asymptotics for certain sums involving sieve weights and the von Mangoldt function; they are established in the 'Multidimensional Selberg sieves' section. Theorems 22,24,26, and 28 use the asymptotics established in Theorems 19 and 20, in combination with Lemma 18, to give various criteria for bounding H m , which all involve finding sufficiently strong candidates for a variety of multidimensional variational problems; these theorems are proven in the 'Reduction to a variational problem' section. These variational problems are analysed in the asymptotic regime of large k in the 'Asymptotic analysis' section, and for small and medium k in the 'The case of small and medium dimension' section, with the results collected in Theorems 23,25,27,and 29.…”

Section: Organization Of the Papermentioning

confidence: 99%

Variants of the Selberg sieve, and bounded intervals containing many primes

Polymath¹

2014

Res Math Sci

130

172

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For any m ≥ 1, let H m denote the quantity lim inf n→∞ (p n+m − p n ). A celebrated recent result of Zhang showed the finiteness of H 1 , with the explicit bound H 1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H 1 ≤ 4680, and then by Maynard to H 1 ≤ 600, who also established for the first time a finiteness result for H m for m ≥ 2, and specifically that H m m 3 e 4m . If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H 1 ≤ 12, improving upon the previous bound H 1 ≤ 16 of Goldston, Pintz, and Yıldırım, as well as the bound H m m 3 e 2m . 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 H 1 ≤ 246 unconditionally and H 1 ≤ 6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h 1 , h 2 , h 3 ), there are infinitely many n for which at least two of n + h 1 , n + h 2 , n + h 3 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 H 1 ≤ 6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound

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Section: Organization Of the Papermentioning

confidence: 99%

Variants of the Selberg sieve, and bounded intervals containing many primes

Polymath¹

2014

Res Math Sci

130

172

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“…When (a, q) = 1, let E(x; q, a) be defined by the relation This result was proved by Bombieri in 1965 [1]. At about the same time, A. I. Vinogradov [28] gave an independent proof of a slightly weaker result. There are numerous proofs of this result available in the literature; see, for example, [4] and [27].…”

Section: Introductionmentioning

confidence: 91%

Small gaps between primes or almost primes

Goldston

Graham

Pintz

et al. 2009

Trans. Amer. Math. Soc.

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Abstract. Let p n denote the n th prime. Goldston, Pintz, and Yıldırım recently proved that lim infWe give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim infIf an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.

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“…Introduction and history. The classical Bombieri-Vinogradov theorem [7,22] touches upon the distribution of primes in arithmetic progressions on average. More precisely, the theorem states the following.…”

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confidence: 99%