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Large gaps between primes. (English) Zbl 1353.11099

After his impressive work on small gaps between primes, this author tackles large gaps between primes achieving significant progress on this problem. The problem investigated is to show that \(g(x)=\sup_{p_n\leq x} (p_{n+1}-p_n)\) is large, where \(p_n\) is the \(n\)th prime. P. Erdős [Duke Math. J. 6, 438–441 (1940; Zbl 0023.29801)], then R. A. Rankin [J. Lond. Math. Soc. 13, 242–247 (1938; Zbl 0019.39403)] showed in the forties that \[ g(x)\geq (c+o(1)) \log x \log\log x \log\log\log\log x (\log\log\log x)^{-2}, \] with \(c=1/3\). All subsequent improvements have been in improving on the value of \(c\). Before this paper, the best value of \(c\) was \(2 e^{\gamma}\), a result of J. Pintz [J. Number Theory 63, No. 2, 286–301 (1997; Zbl 0870.11056)]. In the paper under review, the author shows that \(c\) can be taken to be arbitrarily large by proving that \[ \limsup_{n\to\infty} \frac{p_{n+1}-p_n}{(\log p_n) (\log\log p_n) (\log\log\log\log p_n) (\log\log\log p_n)^{-2}}=\infty. \] He follows closely the Erdős-Rankin construction as explained in the paper by H. Maier and C. Pomerance [Trans. Am. Math. Soc. 322, No. 1, 201–237 (1990; Zbl 0706.11052)] by incorporating sieve ideas from the author’s work on small gaps between primes [Ann. Math. (2) 181, No. 1, 383–413 (2015; Zbl 1306.11073)] as well as from the D. H. J. Polymath project [Res. Math. Sci. 1, Paper No. 12, 83 p. (2014; Zbl 1365.11110); erratum ibid. 2, Paper No. 15, 2 p. (2015)]. The author notes that the same result has been obtained independently by Ford, Green, Konyagin and Tao by incorporating results on linear equations in primes into the same Erdős-Rankin construction.

MSC:

11N05 Distribution of primes
11N36 Applications of sieve methods
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References:

[1] P. Erdös, ”The difference of consecutive primes,” Duke Math. J., vol. 6, pp. 438-441, 1940. · Zbl 0023.29801
[2] K. Ford, B. Green, S. Konyagin, and T. Tao, ”Large gaps between consecutive prime numbers,” Ann. of Math., vol. 184, pp. 935-974, 2016. · Zbl 1338.11083
[3] J. Friedlander and H. Iwaniec, Opera de Cribro, Providence, RI: Amer. Math. Soc., 2010, vol. 57. · Zbl 1226.11099
[4] B. Green and T. Tao, ”Linear equations in primes,” Ann. of Math., vol. 171, iss. 3, pp. 1753-1850, 2010. · Zbl 1242.11071
[5] B. Green and T. Tao, ”The Möbius function is strongly orthogonal to nilsequences,” Ann. of Math., vol. 175, iss. 2, pp. 541-566, 2012. · Zbl 1347.37019
[6] B. Green, T. Tao, and T. Ziegler, ”An inverse theorem for the Gowers \(U^{s+1}[N]\)-norm,” Ann. of Math., vol. 176, iss. 2, pp. 1231-1372, 2012. · Zbl 1282.11007
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[10] J. Pintz, ”Very large gaps between consecutive primes,” J. Number Theory, vol. 63, iss. 2, pp. 286-301, 1997. · Zbl 0870.11056
[11] D. H. J. Polymath, ”Variants of the Selberg sieve, and bounded intervals containing many primes,” Res. Math. Sci., vol. 1, p. 12, 2014. · Zbl 1365.11110
[12] R. A. Rankin, ”The difference between consecutive prime numbers,” J. London Math. Soc., vol. S1-13, iss. 4, p. 242–247, 1938. · Zbl 0019.39403
[13] R. A. Rankin, ”The difference between consecutive prime numbers. V,” Proc. Edinburgh Math. Soc., vol. 13, pp. 331-332, 1962/1963. · Zbl 0121.04705
[14] A. Schönhage, ”Eine Bemerkung zur Konstruktion grosser Primzahllücken,” Arch. Math. \((\)Basel\()\), vol. 14, pp. 29-30, 1963. · Zbl 0108.04504
[15] E. Westzynthius, ”Uber die Verteilung der Zahlen, die zu den \(n\) ersten Primzahlen teilerfremd sind.,” Comm. Phys. Math. Soc. Sci. Fenn., vol. 25, pp. 1-37, 1931. · Zbl 0003.24601
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