2016
DOI: 10.4064/aa8277-11-2015
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A note on small gaps between primes in arithmetic progressions

Abstract: Abstract. We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.

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Cited by 3 publications
(5 citation statements)
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“…This shifting is covered in Section 7. From here, the proof mostly follows the ideas described in [09], albeit with W$W$ a bit bigger than in the original paper. We note that [7] and [2] outline the process for dealing with a larger W$W$, and the effect on the final calculations is minimal.…”
Section: Outlinementioning
confidence: 99%
See 2 more Smart Citations
“…This shifting is covered in Section 7. From here, the proof mostly follows the ideas described in [09], albeit with W$W$ a bit bigger than in the original paper. We note that [7] and [2] outline the process for dealing with a larger W$W$, and the effect on the final calculations is minimal.…”
Section: Outlinementioning
confidence: 99%
“…The only difference here is that W$W$ is of size Xε$X^\epsilon$ for some ε<1r$\epsilon &lt; \frac{1}{r}$. However, such alterations have been addressed in [2, 7], and elsewhere; the only difference is a slight change in the allowable level of distribution.…”
Section: The Maynard–tao Sievementioning
confidence: 99%
See 1 more Smart Citation
“…Although new ideas appear to be required to prove the Twin Prime Conjecture, the breakthroughs of Theorems 7, 8 and 9 have already had several further applications, including new results on large gaps between primes [21,20,52], the resolution of the Erdős discrepancy problem [65], as well as many other results on the distribution of primes [67,51,69,12,6,28,57,42,2,49,3,16,72,70,55,56,33,4,5,1,37,53,58,1,45,43] and correlations of multiplicative functions [68,48,39,38,30,41,27,26].…”
Section: Theorem 9 (Liouville In Short Intervals) For Almost All Inte...mentioning
confidence: 99%
“…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%