Small gaps between primes
TLDR
In this paper, the GPY sieve method for studying prime k-tuples and small gaps between primes was introduced and it was shown that for each k, the prime k -tuples conjecture holds for a positive proportion of admissible k-toples.Abstract:
We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, lim infn(pn+m pn) <1 for every integer m. We also show that lim inf(pn+1 pn) 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 pn) 12 and lim infn(pn+2 pn) 600.read more
Citations
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Journal ArticleDOI
Variants of the Selberg sieve, and bounded intervals containing many primes
TL;DR: In particular, this paper showed that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h 1,n+h 2,h 3 are prime, and also showed 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.
Journal ArticleDOI
New equidistribution estimates of Zhang type
Wouter Castryck,Étienne Fouvry,Gergely Harcos,Emmanuel Kowalski,Philippe Michel,Paul D. Nelson,E. Paldi,János Pintz,Andrew V. Sutherland,Terence Tao,Xiao-Feng Xie +10 more
TL;DR: For arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese remainder theorem conditions, the authors obtained an exponent of distribution 1/2 + 7/300.
Journal ArticleDOI
New equidistribution estimates of Zhang type
TL;DR: For arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, the authors obtained an exponent of distribution for primes of size 1/2 + 7/300.
Posted Content
Dense clusters of primes in subsets
TL;DR: In this article, it was shown that any subset of the primes which is well-distributed in arithmetic progressions contains many primes that are close together, and lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p n+m-p_n\le \epsilon\log{x}$.
Journal ArticleDOI
Long gaps between primes
TL;DR: In this paper, it was shown that max/pn+1 ≤ X (pn + 1 - pn) ≫ log X log log X ε log log log ε/log log log x ε for sufficiently large X.
References
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Bounded gaps between primes
Abstract: It is proved that
lim inf n?8 (p n+1 -p n )<7×10 7 , where p n is the n -th prime.
Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose
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Primes in tuples I
TL;DR: In this article, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
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Limitations to the equi-distribution of primes I
TL;DR: In this article, it was shown that the expected asymptotic formula (x; q; a) (x)==(q) does not hold uniformly in the range q < x= log N x, for any xed N > 0.
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Small gaps between products of two primes
TL;DR: In this paper, it was shown that for any positive integer, if qn is a product of exactly two distinct primes, then (qn+1 - qn) ≤ e −γ(1 + o(1)) infinitely often.
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Higher correlations of divisor sums related to primes III: small gaps between primes
TL;DR: In this paper, it was shown that for any η > 0, a positive proportion of consecutive primes are within 4 + η times the average spacing between primes.