Dense clusters of primes in subsets

Maynard, James

doi:10.1112/s0010437x16007296

Cited by 66 publications

(152 citation statements)

References 19 publications

(61 reference statements)

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“…We may assume that x is suf-http://www.resmathsci.com/content/1/1/12 ficiently large depending on all fixed quantities. By (16), the left-hand side of (29) may be expanded as…”

Section: The Trivial Casementioning

confidence: 99%

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

Polymath¹

2014

Res Math Sci

130

177

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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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“…We may assume that x is suf-http://www.resmathsci.com/content/1/1/12 ficiently large depending on all fixed quantities. By (16), the left-hand side of (29) may be expanded as…”

Section: The Trivial Casementioning

confidence: 99%

“…For the sake of notation, we take i 0 = k, as the other cases are similar. We use (16) to rewrite the left-hand side of (26) as…”

Section: The Elliott-halberstam Casementioning

confidence: 99%

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

Polymath¹

2014

Res Math Sci

130

177

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“…In the direction of these conjectures, the earliest work we found is the paper of Knapowski and Turán (3), who "guess" that the events p n ≡ a ðmod 4Þ and p n+1 ≡ b ðmod 4Þ for the four possibilities of a and b are "not equally probable." However, Knapowski and Turán go on to suggest that πðx; 4; ð1; 1ÞÞ = oðπðxÞÞ, which is now definitively false by Maynard's work (6). The paper (3) was published after the death of both authors, and perhaps they had something else in mind, maybe along the lines of our Conjecture 1.2 above?…”

Section: Significancementioning

confidence: 97%

“…Recent progress in sieve theory has led to a new proof of Shiu's result (see ref. 5), and, moreover, Maynard (6) has shown that πðx; q; ða; . .…”

Section: Introductionmentioning

confidence: 99%

Unexpected biases in the distribution of consecutive primes

Oliver

Soundararajan

2016

Proc. Natl. Acad. Sci. U.S.A.

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Although the sequence of primes is very well distributed in the reduced residue classes ðmod qÞ, the distribution of pairs of consecutive primes among the permissible ϕ(q) 2 pairs of reduced residue classes ðmod qÞ is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy−Littlewood conjectures. The conjectures are then compared with numerical data, and the observed fit is very good.

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“…A wealth of literature has addressed the structure of these sequences, some relevant references include [20][21][22][23][24]. Gaps between primes (ever odd except for the number 2 and 3 the only gap with g = 1)…”

Section: Introductionmentioning

confidence: 99%

On a Dynamical Approach to Some Prime Number Sequences

Lacasa

Luque

Gomez

et al. 2018

Entropy

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Abstract:We show how the cross-disciplinary transfer of techniques from dynamical systems theory to number theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo k are maximally chaotic and, while lacking forbidden patterns, unexpectedly display a non-trivial spectrum of Renyi entropies which suggest that every block of size m > 1, while admissible, occurs with different probability. This non-uniform distribution of blocks for m > 1 contrasts Dirichlet's theorem that guarantees equiprobability for m = 1. We then explore in a similar fashion the sequence of prime gap residues. We numerically find that this sequence is again chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as forbidden patterns emerge for every block of size m > 1. We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy-Littlewood k-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the chaos game to these symbolic sequences, and comparing the Iterated Function System (IFS) attractors found for the experimental sequences with appropriate null models.

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Dense clusters of primes in subsets

Cited by 66 publications

References 19 publications

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

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

Unexpected biases in the distribution of consecutive primes

On a Dynamical Approach to Some Prime Number Sequences

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