Bounded gaps between primes in special sequences

Chua, Lynn; Park, Soo-Hyun; Smith, Geoffrey D.

doi:10.1090/proc/12607

Cited by 12 publications

(13 citation statements)

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“…• If α is an algebraic, irrational number, then by [9] there are infinitely many n such that at least m of [αn], [α(n + 1)], . .…”

Section: The New Resultsmentioning

confidence: 99%

Untitled

2015

Bull. Amer. Math. Soc.

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The infamous twin prime conjecture states that there are infinitely many pairs of distinct primes which differ by 2. Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, making the twin prime conjecture look highly plausible, and the techniques developed help us to better understand other delicate questions about prime numbers that had previously seemed intractable. Zhang even showed that one can take B = 70000000. Moreover, a cooperative team, Polymath8, collaborating only online, had been able to lower the value of B to 4680. They had not only been more careful in several difficult arguments in Zhang's original paper, they had also developed Zhang's techniques to be both more powerful and to allow a much simpler proof (and this forms the basis for the proof presented herein). In November 2013, inspired by Zhang's extraordinary breakthrough, James Maynard dramatically slashed this bound to 600, by a substantially easier method. Both Maynard and Terry Tao, who had independently developed the same idea, were able to extend their proofs to show that for any given integer m ≥ 1 there exists a bound B m such that there are infinitely many intervals of length B m containing at least m distinct primes. We will also prove this much stronger result herein, even showing that one can take B m = e 8m+5. If Zhang's method is combined with the Maynard-Tao setup, then it appears that the bound can be further reduced to 246. If all of these techniques could be pushed to their limit, then we would obtain B(= B 2)= 12 (or arguably to 6), so new ideas are still needed to have a feasible plan for proving the twin prime conjecture. The article will be split into two parts. The first half will introduce the work of Zhang, Polymath8, Maynard and Tao, and explain their arguments that allow them to prove their spectacular results. The second half of this article develops a proof of Zhang's main novel contribution, an estimate for primes in relatively short arithmetic progressions.

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“…• If α is an algebraic, irrational number, then by [9] there are infinitely many n such that at least m of [αn], [α(n + 1)], . .…”

Section: The New Resultsmentioning

confidence: 99%

Untitled

2015

Bull. Amer. Math. Soc.

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“…Subsequent to these works, a number of interesting subsets of the primes have also been shown to exhibit bounded gaps. See [1], [30] for primes in short intervals, [35], [30] for work on Chebotarev sets, and [2], [6] for work on primes in Beatty sequences ( αn + β ) n≥1 . As a consequence of our main theorems, we are able to add to this list that the primes lying in a nil-Bohr set exhibit bounded gaps.…”

Section: Bounded Gaps Between Primes In Bohr Setsmentioning

confidence: 99%

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Shao¹,

Teräväinen²

2021

discrete_Anal

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We establish results of Bombieri-Vinogradov type for the von Mangoldt function Λ(n) twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase e(P(n)); the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes p obeying a "nil-Bohr set" condition, such as α p k < ε, exhibit bounded gaps. Secondly, we show that the Chen primes are well-distributed in nil-Bohr sets, generalizing a result of Matomäki. Thirdly, we generalize the Green-Tao result on linear equations in the primes to primes belonging to an arithmetic progression to large modulus q ≤ x θ , for almost all q.

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“…Subsequent to these works, a number of interesting subsets of the primes have also been shown to exhibit bounded gaps. See [1], [27] for primes in short intervals, [31], [27] for work on Chebotarev sets, and [2], [5] for work on primes in Beatty sequences ( αn + β ) n≥1 .…”

Section: Bounded Gaps Between Primes In Bohr Setsmentioning

confidence: 99%

The Bombieri-Vinogradov theorem for nilsequences

Shao,

Teräväinen

2020

Preprint

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We establish results of Bombieri-Vinogradov type for the von Mangoldt function Λ(n) twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase e(P (n)); the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes p obeying a "nil-Bohr set" condition, such as αp k < ε, exhibit bounded gaps. Secondly, we show that the Chen primes are well-distributed in nil-Bohr sets, generalizing a result of Matomäki. Thirdly, we generalize the Green-Tao result on linear equations in the primes to primes belonging to an arithmetic progression to large modulus q ≤ x θ , for almost all q.

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Bounded gaps between primes in special sequences

Cited by 12 publications

References 11 publications

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The Bombieri-Vinogradov theorem for nilsequences

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