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Primes in intervals of bounded length. (English) Zbl 1319.11064

This survey gives a comprehensive account of the work of Y. Zhang [Ann. Math. (2) 179, No. 3, 1121–1174 (2014; Zbl 1290.11128)]on bounded gaps between primes, and of the subsequent developments by J. Maynard [Ann. Math. (2) 181, No. 1, 383-413 (2015; Zbl 1306.11073)] and Tao. The survey starts with the prime \(k\)-tuples conjecture, before giving Zhang’s theorem, that one has \(p_{n+1}-p_n\leq 7\times 10^7\) for infinitely many \(n\). This leads on to the Maynard-Tao theorem, that \(\liminf_{n\rightarrow\infty}p_{n+m}-p_n\) is finite for any fixed \(m\). There is also a description of the improvements due to the Polymath team. The introduction concludes with a description of earlier work, particularly the key approach developed by D. A. Goldston, J. Pintz and C. Y. Yıldırım [Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096)].
The next few parts of the paper give an account of basic prime number theory, including questions about the distribution in arithmetic progressions. These lead on to the Goldston-Pintz-Yıldırım argument, for which three distinct approaches are given for the estimation of the necessary arithmetic sums.
This is followed by two sections describing how firstly Zhang, and then Maynard and Tao, adapted the GPY method. The Maynard-Tao approach gives rise to a multi-dimensional optimization problem, and the survey gives an account of various different ways in which this can be tackled.
The second chapter of the survey gives a detailed description of the process by which Zhang extended the Bombieri-Vinogradov theorem, for moduli which are suitably smooth. There is an account of the various sharpenings found by the Polymath team, and the survey ends with a description of a range of corollaries and variants of the Maynard-Tao results.
As an epilogue there is a late addition concerning the announcement by Ford, Green, Konyagin and Tao, and independently by Maynard, that they have established an improvement of the old result of Rankin on long gaps between primes. Maynard’s approach employs some of the same ideas that he uses to handle small gaps.

MSC:

11N05 Distribution of primes
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11N13 Primes in congruence classes
11N36 Applications of sieve methods
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References:

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