On some conjectures in additive number theory (Q482499)

The author gives a short panorama of results regarding small gaps between primes and related problems in additive number theory, starting with the Cramer model and arriving at the nowadays astonishing result by \textit{Y. Zhang} on bounded gaps (see [Ann. Math. (2) 179, No. 3, 1121--1174 (2014; Zbl 1290.11128)]); at the time of writing the paper, Maynard had not yet his best constant regarding this: the author gives references, especially for the Polymath project). The arguments exposed comprise, for example (not needing completeness in this short exposition), \textit{H. Maier}'s result [Mich. Math. J. 32, 221--225 (1985; Zbl 0569.10023)], that proves inadequateness of Cramer model; Goldston, Yıldırım and Pintz studies (and Elliott-Halberstam conjecture, as linked to these) and Hardy-Littlewood conjecture for the primes differing by \(2k\) (the \(2k\)-twins conjecture) like, for polynomial prime values, Schinzel hypothesis H [\textit{A. Schinzel} and \textit{W. Sierpiński}, Acta Arith. 4, 185--208 (1958; Zbl 0082.25802)], that he quotes in a section about particular arguments, like Andrica and Rassias conjectures, Germain primes and Cunningham chains.