Maynard, James Small gaps between primes. (English) Zbl 1306.11073 Ann. Math. (2) 181, No. 1, 383-413 (2015). The prime \(k\)-tuples and small gaps between prime numbers are considered. Using a refinement of the Goldston-Pintz-Yildirim sieve method [D. A. Goldston et al., Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096)] the author proves, for instance, the following estimates \[ \liminf_{n\to\infty}\,(p_{n+m}-p_n)\ll m^3\text{{e}}^{4m}, \quad \liminf_{n\to\infty}\,(p_{n+1}-p_n)\leq 600 \] with an absolute constant in sign \(\ll\). Here \(m\) is a natural number, and \(p_{\,l}\) denote the \(l\)-th prime number. Reviewer: Jonas Šiaulys (Vilnius) Cited in 22 ReviewsCited in 107 Documents MSC: 11N05 Distribution of primes 11N36 Applications of sieve methods Keywords:prime number; small gap; sieve method; \(k\)-tuples conjecture; admissible set; Selberg sieve; symmetric polynomial; symmetric matrix Citations:Zbl 1207.11096 PDF BibTeX XML Cite \textit{J. Maynard}, Ann. Math. (2) 181, No. 1, 383--413 (2015; Zbl 1306.11073) Full Text: DOI arXiv OpenURL Online Encyclopedia of Integer Sequences: Twin primes. a(n)-th and (a(n)+1)-st primes are the first pair of primes that differ by exactly 2n; a(n) = -1 if no such pair of primes exists. p and p+12 are both prime. Primes p such that p + 600 is also prime. Small gaps between primes - refinement of the GPY sieve method. Related to small gaps between primes: a(n) = A235686(n)/2. References: [1] P. D. T. A. Elliott and H. Halberstam, ”A conjecture in prime number theory,” in Symposia Mathematica, Vol. IV, London: Academic Press, 1970, pp. 59-72. · Zbl 0238.10030 [2] J. Friedlander and A. Granville, ”Limitations to the equi-distribution of primes. I,” Ann. of Math., vol. 129, iss. 2, pp. 363-382, 1989. · Zbl 0671.10041 [3] D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim, ”Small gaps between products of two primes,” Proc. Lond. Math. Soc., vol. 98, iss. 3, pp. 741-774, 2009. · Zbl 1213.11171 [4] D. A. Goldston, J. Pintz, and C. Y. Yildirim, ”Primes in tuples. III. On the difference \(p_{n+\nu}-p_n\),” Funct. Approx. Comment. Math., vol. 35, pp. 79-89, 2006. · Zbl 1196.11123 [5] D. A. Goldston, J. Pintz, and C. Y. Yildirim, ”Primes in tuples. I,” Ann. of Math., vol. 170, iss. 2, pp. 819-862, 2009. · Zbl 1207.11096 [6] D. A. Goldston and C. Y. Yildirim, ”Higher correlations of divisor sums related to primes. III. Small gaps between primes,” Proc. Lond. Math. Soc., vol. 95, iss. 3, pp. 653-686, 2007. · Zbl 1118.11040 [7] D. H. J. Polymath, New equidistribution estimates of Zhang type, and bounded gaps between primes. · Zbl 1365.11110 [8] A. Selberg, Collected Papers. Vol. II, New York: Springer-Verlag, 1991. · Zbl 0729.11001 [9] Y. Zhang, ”Bounded gaps between primes,” Ann. of Math., vol. 179, iss. 3, pp. 1121-1174, 2014. · Zbl 1290.11128 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.