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

Polymath, D. H. J.

doi:10.48550/arxiv.1407.4897

Cited by 9 publications

(15 citation statements)

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“…If we fix m, k and η P r0, 1q with M k,η ´4m ą 0 (where M k,η is as in (4.3)), and if we assume the remaining hypotheses of Theorem 4.3 hold (disregarding (4.18)), then we can show, for all N ě Npm, k, ǫq, that |Hpnq X P| ě m `1 for at least one n P pN, 2Ns with n " b mod W . This follows from an essentially identical argument to that presented in [13,17], although there are two differences in our setting that potentially affect the argument. Namely, w is considerably larger here than in [13] or [17] (we take w " ǫ log N instead of w " log 3 N), and the elements of H here may vary with N.…”

Section: 20)mentioning

confidence: 77%

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On limit points of the sequence of normalized prime gaps

Banks

Freiberg

Maynard³

2016

Proc. London Math. Soc.

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Let p n denote the nth smallest prime number, and let L denote the set of limit points of the sequence tpp n`1´pn q{ log p n u 8 n"1 of normalized differences between consecutive primes. We show that for k " 9 and for any sequence of k nonnegative real numbers β 1 ď β 2 ď¨¨¨ď β k , at least one of the numbers β j´βi (1 ď i ă j ď k) belongs to L. It follows that at least 12.5% of all nonnegative real numbers belong to L.

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Section: 20)mentioning

confidence: 77%

“…for some absolute constant c. Moreover, the Polymath project has refined the method of [13] slightly, allowing one to reduce the condition M k ą 2θ ´1m to M k,η ą 2θ ´1m for some 0 ď η ď θ ´1 ´1. They have also [17,Theorem 3.13] produced the bound…”

Section: Outline Of the Proofmentioning

confidence: 99%

On limit points of the sequence of normalized prime gaps

Banks

Freiberg

Maynard³

2016

Proc. London Math. Soc.

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“…Another polymath project [8, Theorem 3.2] has since refined Maynard's work so that one can take k 2 " 50 and k m " ce p4´28{157qm . (In [5,8], only tuples of monic linear forms are treated explicitly, although the results should extend to general linear forms as considered in [3]. )…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Consecutive primes in tuples

2015

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In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple Hpxq " tgx`h j u k j"1 of linear forms in Zrxs, the set Hpnq " tgn`h j u k j"1 contains at least m primes for infinitely many n P N. In this note, we deduce that Hpnq " tgn`h j u k j"1 contains at least m consecutive primes for infinitely many n P N. We answer an old question of Erdős and Turán by producing strings of m`1 consecutive primes whose successive gaps δ 1 , . . . , δ m form an increasing (resp. decreasing) sequence. We also show that such strings exist with δ j´1 | δ j for 2 ď j ď m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D.

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“…where the supremum is taken over F described above with I k (F ) = 0, J This bound is strengthened slightly in [19] to (5.1) M k ≥ log k + O(1).…”

Section: Proof Of Theoremmentioning

confidence: 99%