Bounded gaps between primes in number fields and function fields

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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“…From (12) and the crucial condition (15), it follows that N > 0 if x is sufficiently large. On the other hand, the sum…”

Section: Outline Of the Key Ingredientsmentioning

confidence: 99%

“…where B was defined in (12), and Such asymptotics are standard in the literature (see, e.g. [37] for some similar computations).…”

Section: Multidimensional Selberg Sievesmentioning

confidence: 99%

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

Polymath¹

2014

Res Math Sci

127

172

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“…The fact that one can restrict the entire argument to an arithmetic progression also allows one to get some control on the joint distribution of various arithmetic functions. There have been many recent works making use of these flexibilities in the setup of the sieve method, including [58,13,7,21,48,34,3,39,4,14,61,59,46,47,28,5,6,1,32,43,49].…”

Section: Other Applications and Further Readingmentioning

confidence: 99%

Gaps Between Primes

Maynard

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…Although we will eventually need to fix w in the proof of Theorem 2, it is helpful to think of w tending to infinity up until that point, and we use the asymptotic notation o(1) for a quantity tending to zero as w → ∞. Since H is admissible, we can take w > deg(h k ) and fix a congruence class b (mod W ) with (W, b + h j ) = 1 for each h j ∈ H. We allow n → ∞, and we always insist that w ≪ log log n so that deg(W ) ≪ log n and terms that tend to zero as n → ∞ are also o (1).…”

Section: Setupmentioning

confidence: 99%

“…These differ from the integer weights of Maynard in [6], but produce essentially the same results. A version of Proposition 1 for Maynard's weights in F q [t] appeared in [1], but it is unclear how to adapt their setup to obtain a concentration estimate comparable to Proposition 2. Such an estimate was the heart of Pintz' argument in [7] (see also [9,Proposition 14]).…”

Section: The Density Argumentmentioning

confidence: 99%