Abstract
This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang [18] had revived in us an intention to produce a second edition of our book “Opera de Cribro”, one which would include an account of Zhang’s result, stressing the sieve aspects of the method. A complete and connected version of the proof, in our style but not intended for journal publication, seemed a natural first step in this project.
Following the further spectacular advance given by J. Maynard (arXiv:1311. 4600, Nov 20, 2013), we have had to re-think our position. Maynard’s method, at least in its current form, proceeds from GPY in quite a different direction than does Zhang’s, and achieves numerically superior results. Consequently, although Zhang’s contribution to the distribution of primes in arithmetic progressions certainly retains its importance, the fact remains that much of the material in this paper would no longer appear in a new edition of our book. Because this paper contains some innovations that we do not wish to become lost, we have decided to make the work publicly available in its original format. We are extremely pleased to be able to include it in the current volume, commemorating the one hundred and twenty-fifth anninversary of the birth of Srinivasa Ramanujan.
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References
E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A 293 (1966), 1–18.
E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), 203–251.
E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli. II, Math. Ann. 277 (1987), 361–393.
E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli. III, J. Amer. Math. Soc. 2 (1989), 215–224.
P. Erdős, The difference of consecutive primes, Duke Math. J. 6 (1940), 438–441.
E. Fouvry, Autour du théorème de Bombieri-Vinogradov, Acta Math. 152 (1984), 219–244.
E. Fouvry and H. Iwaniec, Primes in arithmetic progressions, Acta Arith. 42 (1983), 197–218.
E. Fouvry, E. Kowalski, and P. Michel, Algebraic twists of modular forms and Hecke orbits (2012), arXiv:1207.0617.
E. Fouvry, E. Kowalski, and P. Michel, On the exponent of distribution of the ternary divisor function (2013), arXiv:1304.3199.
J. B. Friedlander and H. Iwaniec, Incomplete Kloosterman sums and a divisor problem, Ann. of Math. (2) 121 (1985), 319–350.
J. B. Friedlander and H. Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010.
D. A. Goldston and C. Y. Yildirim, Higher correlations of divisor sums related to primes. III. Small gaps between primes, Proc. Lond. Math. Soc. (3) 95 (2007), 653–686.
D. A. Goldston, J. Pintz, and C. Y. Yildirim, Primes in tuples. I, Ann. of Math. (2) 170 (2009), 819–862.
M. N. Huxley, On the differences of primes in arithmetical progressions, Acta Arith. 15 (1968/1969), 367–392.
Ju. V. Linnik, The dispersion method in binary additive problems, American Mathematical Society, Providence, R.I., 1963.
H. Maier, Small differences between prime numbers, Michigan Math. J. 35 (1988), 323–344.
Y. Motohashi and J. Pintz, A smoothed GPY sieve, Bull. Lond. Math. Soc. 40 (2008), 298–310.
Y. Zhang, Bounded gaps between primes, Ann. Math 179 (2014), 1121–1174.
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Research of JF supported in part by NSERC Grant A5123 and that of HI supported in part by NSF Grant DMS-1101574. We thank Leo Goldmakher and Pedro Pontes for their help with the physical appearance of the paper.
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Friedlander, J.B., Iwaniec, H. Close encounters among the primes. Indian J Pure Appl Math 45, 633–689 (2014). https://doi.org/10.1007/s13226-014-0083-6
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DOI: https://doi.org/10.1007/s13226-014-0083-6