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The analytic principle of the large sieve


Author: Hugh L. Montgomery
Journal: Bull. Amer. Math. Soc. 84 (1978), 547-567
MSC (1970): Primary 10H30
DOI: https://doi.org/10.1090/S0002-9904-1978-14497-8
MathSciNet review: 0466048
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  • 1. R. A. Adams, Sovolev spaces, Academic Press, New York, 1975. MR 450957
  • 2. M. B. Barban, The density of zeros of Dirichlet L-series and the problem of the addition of primes and almost primes, Dokl. Akad. Nauk UzSSR 1 (1963), 9-10. (Russian)
  • 3. M. B. Barban, The "density" of the zeros of Dirichlet L-series and the problem of the sum of primes and "near primes", Mat. Sb. 61 (103) (1963), 418-425. MR 171765
  • 4. M. B. Barban, The "large sieve" method and its applications in the theory of numbers, Uspehi Mat. Nauk 21 (1966), 51-102 = Russian Math. Surveys 21 (1966), 49-103. MR 199171
  • 5. R. Bellman, Almost orthogonal series, Bull. Amer. Math. Soc. 50 (1944), 517-519. MR 10639
  • 6. R. P. Boas, A general moment problem, Amer. J. Math. 63 (1941), 361-370. MR 3848
  • 7. E. Bombieri, On the large sieve, Mathematika 12 (1965), 201-225. MR 197425
  • 8. E. Bombieri, Nuovi metodi e nuovi risultati nella teoria dei numeri, Bol. Un. Mat. Ital. (4) 1 (1968), 96-106. MR 234928
  • 9. E. Bombieri, On a theorem of van Lint and Richert, Symposia Mathematica. IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 175-180. MR 279065
  • 10. E. Bombieri, A note on the large sieve, Acta Arith. 18 (1971), 401-404. MR 286773
  • 11. E. Bombieri, On the large sieve inequalities and their applications, Proc. Internat. Conf. Number Theory (Moscow, 1971), Trudy Mat. Inst. Steklov 132 (1973), 251-256, 266. MR 404176
  • 12. E. Bombieri, Le grand crible dans la théorie analytique des nombres, Astérisque 18, Société Math. France, 1974. MR 371840
  • 13. E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A 293 (1966), 1-18. MR 199165
  • 14. E. Bombieri and H. Davenport, On the large sieve method, Number Theory and Analysis (Papers in honor of Edmund Landau), Plenum, New York, 1969, pp. 9-22. MR 260703
  • 15. E. Bombieri and H. Davenport, Some inequalities involving trigonometrical polynomials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1969), 223-241. MR 249391
  • 16. D. A. Burgess, The average of the least primitive root modulo p2, Acta Arith. 18 (1971), 263-271. MR 291118
  • 17. J. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Kexue Tongbao 17 (1966), 385-386. MR 207668
  • 18. J. Chen, On the representation of a large even integer as a sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157-176. MR 434997
  • 19. H. Davenport, Multiplicative number theory, Markham, Chicago, 1967. (Delete Theorems 4, 4A in §23.) MR 217022
  • 20. H. Davenport, The zeros of trigonometrical polynomials, Mathematika 19 (1972), 88-90. MR 316953
  • 21. H. Davenport and H. Halberstam, The values of a trigonometric polynomial at well spaced points, Mathematika 13 (1966), 91-96. Corrigendum and addendum, Mathematika 14 (1967), 229-232. MR 197427
  • 22. H. Davenport and H. Halberstam, Primes in arithmetic progressions, Michigan Math. J. 13 (1966), 485-489. Corrigendum, Michigan Math. J. 15 (1968), 505. MR 200257
  • 23. P. D. T. A. Elliott, The Turán-Kubilius inequality, and a limitation theorem for the large sieve, Amer. J. Math. 92 (1970), 293-300. MR 263760
  • 24. P. D. T. A. Elliott, On inequalities of large sieve type, Acta Arith. 18 (1971), 405-422 MR 286774
  • 25. P. D. T. A. Elliott, On connections between the Turán-Kubilius inequality and the large sieve: some applications, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 77-82. MR 344215
  • 26. P. Erdős, Remarks on number theory. V, Mat. Lapok 17 (1966), 135-155. MR 217038
  • 27. P. Erdős and A. Rényi, Some remarks on the large sieve of Yu. V. Linnik, Ann. Univ. Sci. Budapest. Eőtvős Sect. Math. 11 (1968), 3-13. MR 241378
  • 28. M. Forti and C. Viola, On the large sieve type estimates for the Dirichlet series operator, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 31-49. MR 506095
  • 29. P. X. Gallagher, The large sieve, Mathematika 14 (1967), 14-20. MR 214562
  • 30. P. X. Gallagher, Bombieri's mean value theorem, Mathematika 15 (1968), 1-6. MR 237442
  • 31. P. X. Gallagher, A large sieve density estimate near σ = 1, Invent. Math. 11 (1970), 329-339. MR 279049
  • 32. P. X. Gallagher, Sieving by prime powers, Acta Arith 24 (1973), 491-497. MR 337844
  • 33. P. X. Gallagher, The large sieve and probabilistic Galois theory, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 91-101. MR 332694
  • 34. S. W. Graham, Applications of sieve methods, Ph.D. Dissertation, Univ. of Michigan, Ann Arbor, 1977.
  • 35. H. Halberstam, The large sieve, Number Theory (Colloq. János Bolyai Math. Soc., Debrecen, 1968), North-Holland, Amsterdam, 1970, pp. 123-131. MR 272744
  • 36. H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974. MR 424730
  • 37. H. Halberstam and K. F. Roth, Sequences, Oxford Univ. Press, London and New York, 1966. MR 210679
  • 38. E. Hlawka, Bemerkungen zum grossen Sieb von Linnik, Österreich. Akad. Wiss. Lit. Mainz Abh. Math.-Natur. Kl. S.-B. II 178 (1970), 13-18. MR 265314
  • 39. E. Hlawka, Zum grossen Sieb von Linnik, Acta Arith. 27 (1975), 89-100. MR 366845
  • 40. C. Hooley, On the Barban-Davenport-Halberstam theorem. I, J. Reine Angew. Math. 274/275 (1975), 206-223. MR 382202
  • 41. M. N. Huxley, The large sieve inequality for algebraic number fields, Mathematika 15 (1968), 178-187. MR 237455
  • 42. M. N. Huxley, The large sieve inequality for algebraic number fields. II. Means of moments of Hecke zeta-functions, Proc. London Math. Soc. (3) 21 (1970), 108-128. MR 271061
  • 43. M. N. Huxley, The large sieve inequality for algebraic number fields. III. Zero-density results, J. London Math. Soc. (2) 3 (1971), 233-240. MR 276196
  • 44. M. N. Huxley, The distribution of prime numbers, Oxford Mathematical Monographs, Oxford Univ. Press, London and New York, 1972. MR 444593
  • 45. M. N. Huxley, Irregularity in sifted sequences, J. Number Theory 4 (1972), 437-454. MR 311618
  • 46. A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), no. 1, 367–379. MR 1545625, https://doi.org/10.1007/BF01180426
  • 47. J. Johnsen, On the large sieve method in GF[q, x], Mathematika 18 (1971), 172-184. MR 302617
  • 48. I. Kobayashi, Remarks on the large sieve method, Proc. United States-Japan Seminar on Number Theory, Tokyo, 1971. MR 382206
  • 49. I. Kobayashi, A note on the Selberg sieve and the large sieve, Proc. Japan Acad. 49 (1973), 1-5. MR 325556
  • 50. I. P. Kubilius, Probabilistic methods in the theory of numbers, Uspehi Mat. Nauk 11 (1956), no. 2 (68), 31-36. MR 79025
  • 51. Ju. V. Linnik, The large sieve, Dokl. Akad. Nauk SSSR 30 (1941), 292-294. (Russian) MR 4266
  • 52. Ju. V. Linnik, A remark on the least quadratic non-residue, Dokl. Akad. Nauk SSSR 36 (1941), 119-120. (Russian) MR 7758
  • 53. J. Marcinkiewicz and A. Zygmund, Proof of a gap theorem, Duke Math. J. 4 (1938), no. 3, 469–472. MR 1546068, https://doi.org/10.1215/S0012-7094-38-00439-9
  • 54. K. R. Matthews, On a bilinear form associated with the large sieve, J. London Math. Soc. 5 (1972), 567-570. MR 318082
  • 55. K. R. Matthews, On an inequality of Davenport and Halberstam, J. London Math. Soc. 4 (1972), 638-642. MR 302582
  • 56. K. R. Matthews, Hermitian forms and the large and small sieves, J. Number Theory 5 (1973), 16-23. MR 321895
  • 57. Ming-Chit Liu, On a result of Davenport and Halberstam, J. Number Theory 1 (1969), 385-389. MR 249392
  • 58. H. L. Montgomery, A note on the large sieve, J. London Math. Soc. 43 (1968), 93-98. MR 224585
  • 59. H. L. Montgomery, Mean and large values of Dirichlet polynomials, Invent. Math. 8 (1969), 334-345. MR 268130
  • 60. H. L. Montgomery, Zeros of L-functions, Invent. Math. 8 (1969), 346-354. MR 249375
  • 61. H. L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Math., vol. 227, Springer-Verlag, Berlin, 1971. MR 337847
  • 62. H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134. MR 374060
  • 63. H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc. (2) 8 (1974), 73-81. MR 337775
  • 64. H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353-370. MR 374063
  • 65. Y. Motohashi, A note on the large sieve, Proc. Japan Acad. 53 (1977), 17-19. MR 432571
  • 66. Y. Motohashi, On Gallagher's prime number theorem, Proc. Japan Acad. 53 (1977), 50-52. MR 480383
  • 67. Y. Motohashi, Introduction to the theory of the distribution of prime numbers, Sûgaku 26 (1974), no.1, 1-12. MR 414501
  • 68. Y. Motohashi, On the density theorem of Linnik, Proc. Japan Acad. 51 (1975), suppl. 815-917. MR 401677
  • 69. Y. Motohashi, A note on the large sieve. II. Proc. Japan Acad. 53 (1977), 122-124. MR 485749
  • 70. Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142
  • 71. P. A. B. Pleasants, A sum related to the distribution modulo1 of sets of real numbers, Quart. J. Math. Oxford Ser. (2) 21 (1970), 321-336. MR 271062
  • 72. A. Rényi, On the representation of an even number as the sum of a single prime and a single almost-prime number, Dokl. Akad. Nauk SSSR 56 (1947), 455-458. (Russian) MR 21958
  • 73. A. Rényi, On the representation of an even number as the sum of a single prime and a single almost-prime number, Izv. Akad. Nauk. SSSR Ser. Mat. 12 (1948), 57-78; English transl., Amer. Math. Soc. Transl. (2) 19 (1962), 299-321. MR 131413
  • 74. A. Rényi, Un nouveau théorèm concernant les fonctions indépendantes et ses applications à la théorie des nombres, J. Math. Pures Appl. (9) 28 (1949), 137-149. MR 31505
  • 75. A. Rényi, Probability methods in number theory, Publ. Math. Collectae Budapest 1 (1949), no. 21, 1-9. MR 36782
  • 76. A. Rényi, On a theorem of the theory of probability and its application in number theory, Časopis Pešt. Mat. Fys. 74 (1949), 167-175. MR 39750
  • 77. A. Rényi, Sur un théorème général de probabilité, Ann. Inst. Fourier (Grenoble) 1 (1950), 43-52. MR 54187
  • 78. A. Rényi, On the large sieve of Ju. V. Linnik, Compositio Math 8 (1950), 68-75. MR 34407
  • 79. A. Rényi, On the probabilistic generalization of the large sieve of Linnik, Magyar Tud. Akad. Mat. Kutató Int. Kőzl. 3 (1958), 199-206. MR 111073
  • 80. A. Rényi, Probabilistic methods in number theory, Proc. Internat. Congress Math., Cambridge Univ. Press, London and New York, 1958, pp. 529-539. MR 118707
  • 81. A. Rényi, A new version of the probabilistic generalization of the large sieve, Acta Math. Acad. Sci. Hungar 10 (1959), 217-226. MR 111074
  • 82. G. J. Rieger, Zum Sieb von Linnik, Arch. Math. 11 (1960), 14-22. MR 153632
  • 83. G. J. Rieger, Das grosse Sieb von Linnik für algebraische Zahlen, Arch. Math. 12 (1961), 184-187. MR 133318
  • 84. P. M. Ross, On Chen's theorem that each large even number has the form p1+ p2 or p1+ p2p3, J. London Math. Soc. 10 (1975), 500-506. MR 389816
  • 85. K. F. Roth, A remark on integer sequences, Acta Arith. 9 (1964), 257-260. MR 168545
  • 86. K. F. Roth, On the large sieves of Linnik and Rényi, Mathematika 12 (1965), 1-9. MR 197424
  • 87. K. F. Roth, The large sieve. Inaugural Lecture, January 23, 1968, Imperial College of Science and Technology, London, 1968. MR 237458
  • 88. A. G. Samandarov, On the large sieve in algebraic number fields, Mat. Zametki 6 (1967), 673-680. MR 223331
  • 89. W. Schaal, On the large sieve method in algebraic number fields, J. Number Theory 2 (1970), 249-270. MR 272745
  • 90. I. Schur, Bemerkungen zur Theorie der beschrankten Bilinearformen mit unendlich vielen Verändlichen, J. Reine Angew. Math. 140 (1911), 1-28.
  • 91. W. Schwarz, Einführung in Siebmethoden der analytischen Zahlentheorie, Bibliographisches Institut, Mannheim-Vienna-Zurich, 1974. MR 409392
  • 92. S. L. Sobolev, Applications of functional analysis in mathematical physics, Transl. Math. Monographs, vol. 7, 1963. MR 165337
  • 93. A. V. Sokolovskiĭ, The large sieve, Acta Arith. 25 (1973/74), 301-306. MR 347755
  • 94. E. C. Titchmarsh, A class of trigonometrical series, J. London Math. Soc. 3 (1928), 300-304.
  • 95. S. Uchiyama, The maximal large sieve, Hokkaido Math. J. 1 (1972), 117-126. MR 321896
  • 96. A. I. Vinogradov, On the density hypothesis for Dirichlet L-functions, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903-934. Correction: Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 719-720.
  • 97. Norbert Wiener, A class of gap theorems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 3 (1934), no. 3-4, 367–372. MR 1556735
  • 98. R. J. Wilson, The large sieve in algebraic number fields, Mathematika 16 (1969), 189-204. MR 263774
  • 99. D. Wolke, Farey-Bruche mit primen Nenner und das grosse Sieb, Math. Z. 114 (1970), 145-158. MR 260704
  • 100. D. Wolke, Einige Anwendungen des grossen Siebes auf zahlentheoretische Funktionen, Habilitationsschrift, Phillipps-Universitat Marburg/Lahn, Marburg/Lahn, 1970. MR 279058
  • 101. D. Wolke, On the large sieve with primes, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239-247. MR 291121
  • 102. D. Wolke, Über eine Ungleichung von A. I. Vinogradov, Arch. Math. (Basel) 23 (1972), 625-629. MR 321886
  • 103. D. Wolke, A lower bound for the large sieve inequality, Bull. London Math. Soc. 6 (1974), 315-318. MR 354587

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DOI: https://doi.org/10.1090/S0002-9904-1978-14497-8

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