New equidistribution estimates of Zhang type

Algebra and Number Theory

Volumn 8, Issue 9, 2014, Pages 2067-2199

  Castryck, Wouter ^a   Fouvry, Étienne ^b   Harcos, Gergely ^c   Kowalski, Emmanuel ^d   Michel, Philippe ^e   Nelson, Paul ^d   Paldi, Eytan ^f   Pintz, János ^c   Sutherland, Andrew V ^g   Tao, Terence ^h   Xie, Xiao Feng ⁱ  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b UNIVERSITÉ PARIS SACLAY   (France)

^c ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS   (Hungary)

^d ETH ZURICH   (Switzerland)

^e EPFL   (Switzerland)

^f TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^g MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^h UNIVERSITY OF CALIFORNIA   (United States)

ⁱ CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Bombieri Vinogradov theorem; Elliott Halberstam conjecture; Prime gaps

Indexed keywords

EID: 84928128898     PISSN: 19370652     EISSN: None     Source Type: Journal    
DOI: 10.2140/ant.2014.8.2067     Document Type: Article

References (52)

1. M. B. Barban and P. P. Vehov, “Summation of multiplicative functions of polynomials”, Mat. Zametki 5 (1969), 669–680. MR 40 #4221 Zbl 0192.39103
2. E. Bombieri, “Counting points on curves over finite fields (d’après S. A. Stepanov)” exposé no. 430, 234–241 in Séminaire Bourbaki, 1972/1973, Lecture Notes in Math. 383, Springer,Berlin, 1974. MR 55 #2912 Zbl 0307.14011
3. E. Bombieri, J. B. Friedlander, and H. Iwaniec, “Primes in arithmetic progressions to large moduli”, Acta Math. 156:3-4 (1986), 203–251. MR 88b:11058 Zbl 0588.10042
4. E. Bombieri, J. B. Friedlander, and H. Iwaniec, “Primes in arithmetic progressions to large moduli, II”, Math. Ann. 277:3 (1987), 361–393. MR 88f:11085 Zbl 0625.10036
5. E. Bombieri, J. B. Friedlander, and H. Iwaniec, “Primes in arithmetic progressions to large moduli, III”, J. Amer. Math. Soc. 2:2 (1989), 215–224. MR 89m:11087 Zbl 0674.10036
6. T. Cochrane and C. Pinner, “Using Stepanov’s method for exponential sums involving rational functions”, J. Number Theory 116:2 (2006), 270–292. MR 2006j:11113 Zbl 1093.11058
7. P. Deligne, “La conjecture de Weil, I”, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307. MR 49 #5013 Zbl 0287.14001
8. P. Deligne, “La conjecture de Weil, II”, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252. MR 83c:14017 Zbl 0456.14014
9. É. Fouvry, “Autour du théorème de Bombieri–Vinogradov”, Acta Math. 152:3-4 (1984), 219–244. MR 85m:11052 Zbl 0552.10024
10. É. Fouvry, “Sur le problème des diviseurs de Titchmarsh”, J. Reine Angew. Math. 357 (1985), 51–76. MR 87b:11090 Zbl 0547.10039
11. E. Fouvry and H. Iwaniec, “On a theorem of Bombieri–Vinogradov type”, Mathematika 27:2 (1980), 135–152. MR 82h:10057 Zbl 0469.10027
12. E. Fouvry and H. Iwaniec, “Primes in arithmetic progressions”, Acta Arith. 42:2 (1983), 197–218. MR 84k:10035 Zbl 0517.10045
13. É. Fouvry and H. Iwaniec, “The divisor function over arithmetic progressions”, Acta Arith. 61:3 (1992), 271–287. MR 93g:11089 Zbl 0764.11040
14. É. Fouvry, E. Kowalski, and P. Michel, “An inverse theorem for Gowers norms of trace functions over Fp”, Math. Proc. Cambridge Philos. Soc. 155:2 (2013), 277–295. MR 3091520 Zbl 06203760
15. E. Fouvry, E. Kowalski, and P. Michel, “On the conductor of cohomological transforms”, preprint, 2013. arXiv 1310.3603
16. E. Fouvry, E. Kowalski, and P. Michel, “The sliding-sum method for short exponential sums”, preprint, 2013. arXiv 1307.0135
17. E. Fouvry, E. Kowalski, and P. Michel, “Algebraic twists of modular forms and Hecke orbits”, preprint, 2014. arXiv 1207.0617
18. E. Fouvry, E. Kowalski, and P. Michel, “On the exponent of distribution of the ternary divisor function”, Mathematika (online publication June 2014).
19. E. Fouvry, E. Kowalski, and P. Michel, “Trace functions over finite fields and their applications”, preprint, 2014, http://www.math.ethz.ch/∼kowalski/trace-functions pisa.pdf. To appear in “Colloquium De Giorgi 2013 and 2014”.
20. J. B. Friedlander and H. Iwaniec, “Incomplete Kloosterman sums and a divisor problem”, Ann. of Math. 2/ 121:2 (1985), 319–350. MR 86i:11050 Zbl 0572.10029
21. P. X. Gallagher, “Bombieri’s mean value theorem”, Mathematika 15 (1968), 1–6. MR 38 #5724 Zbl 0174.08103
22. D. A. Goldston, J. Pintz, and C. Y. Yıldırım, “Primes in tuples, I”, Ann. Of Math..2/ 170:2 (2009), 819–862. MR 2011c:11146 Zbl 1207.11096
23. S. W. Graham and C. J. Ringrose, “Lower bounds for least quadratic nonresidues”, pp. 269–309 in Analytic number theory (Allerton Park, IL, 1989), edited by B. C. Berndt et al., Progr. Math. 85, Birkhäuser, Boston, 1990. MR 92d:11108 Zbl 0719.11006
24. D. R. Heath-Brown, “Hybrid bounds for Dirichlet L-functions”, Invent. Math. 47:2 (1978), 149–170. MR 58 #5549 Zbl 0362.10035
25. D. R. Heath-Brown, “Prime numbers in short intervals and a generalized Vaughan identity”, Canad. J. Math. 34:6 (1982), 1365–1377. MR 84g:10075 Zbl 0478.10024
26. D. R. Heath-Brown, “The divisor function d3.n/ in arithmetic progressions”, Acta Arith. 47:1 (1986), 29–56. MR 88a:11088 Zbl 0549.10034
27. D. R. Heath-Brown, “The largest prime factor of X3 C2”, Proc. London Math. Soc..3/ 82:3 (2001), 554–596. MR 2001m:11158 Zbl 1023.11048
28. H. Iwaniec, “A new form of the error term in the linear sieve”, Acta Arith. 37 (1980), 307–320. MR 82d:10069 Zbl 0444.10038
29. H. Iwaniec and E. Kowalski, Analytic number theory, Americ athematical Society Colloquium Publications 53, American Mathematical Society, Providence, RI, 2004. MR 2005h:11005 Zbl 1059.11001
30. N. M. Katz, Sommes exponentielles, Astérisque 79, Société Mathématique de France, Paris, 1980. MR 82m:10059 Zbl 0469.12007
31. N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies 116, Princeton University Press, 1988. MR 91a:11028 Zbl 0675.14004
32. N. M. Katz, “L-functions and monodromy: four lectures on Weil II”, Adv. Math. 160:1 (2001), 81–132. MR 2002c:11066 Zbl 1016.14011
33. H. D. Kloosterman, “On the representation of numbers in the form ax2 Cby2 C cz2 Cdt2”, Acta Math. 49:3-4 (1927), 407–464. MR 1555249 Zbl 53.0155.01
34. E. Kowalski, “Some aspects and applications of the Riemann hypothesis over finite fields”, Milan J. Math. 78:1 (2010), 179–220. MR 2011g:11229 Zbl 1271.11113
35. G. Laumon, “Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil”, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210. MR 88g:14019 Zbl 0641.14009
36. J. V. Linnik, The dispersion method in binary additive problems, American Mathematical Society, Providence, R.I., 1963. MR 29 #5804 Zbl 0112.27402
37. J. Maynard, “Small gaps between primes”, preprint, 2013. arXiv 1311.4600
38. P. Michel, “Minorations de sommes d’exponentielles”, Duke Math. J. 95:2 (1998), 227–240. MR 99i:11069 Zbl 0958.11056
39. H. L. Montgomery and R. C. Vaughan, Multiplicative numbe theory, I: Classical theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007. MR 2009b:11001 Zbl 1142.11001
40. L. J. Mordell, “On a sum analogous to a Gauss’s sum”, Q. J. Math., Oxf. Ser. 3 (1932), 161–167. Zbl 0005.24603
41. Y. Motohashi, “An induction principle for the generalization of Bombieri’s prime number theorem”, Proc. Japan Acad. 52:6 (1976), 273–275. MR 54 #10171 Zbl 0355.10035
42. Y. Motohashi and J. Pintz, “A smoothed GPY sieve”, Bull. Lond. Math. Soc. 40:2 (2008), 298–310. MR 2009d:11132 Zbl 1278.11090
43. G. I. Perel0muter, “Estimate of a sum along an algebraic curve”, Mat. Zametki 5 (1969), 373–380. In Russian. MR 39 #2764 Zbl 0179.49903
44. J. Pintz, “A note on bounded gaps between primes”, preprint, 2013. arXiv 1306.1497
45. D. H. J. Polymath, “New equidistribution estimates of Zhang type, and bounded gaps between primes”, preprint, 2014. arXiv 1402.0811v2
46. D. H. J. Polymath, “Variants of the Selberg sieve, and bounded intervals containing many primes”, preprint, 2014. arXiv 1407.4897
47. P. Deligne (editor), Cohomologie étale (SGA 412), Lecture Notes in Math. 569, Springer, Berlin, 1977. MR 57 #3132 Zbl 0345.00010
48. P. Shiu, “A Brun–Titchmarsh theorem for multiplicative functions”, J. Reine Angew. Math. 313 (1980), 161–170. MR 81h:10065 Zbl 0412.10030
49. H. Siebert, “Einige Analoga zum Satz von Siegel–Walfisz”, pp. 173–184 in Zahlentheorie (Tagung des Math. Forschungsinst., Oberwolfach, 1970), edited by M. Barner and W. Schwarz, Bibliographisches Inst., Mannheim, 1971. MR 51 #391 Zbl 0221.10041
50. R.-C. Vaughan, “Sommes trigonométriques sur les nombres premiers”, C. R. Acad. Sci. Paris Sér. A-B 285:16 (1977), A981–A983. MR 58 #16555 Zbl 0374.10025
51. A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Actualités Sci. Ind. 1041, Hermann et Cie, Paris, 1948. MR 10,262c Zbl 0036.16001
52. A. Y. Zhang, “Bounded gaps between primes”, Ann. of Math. 2/ 179:3 (2014), 1121–1174. MR 3171761 Zbl 1290.11128