The analytic principle of the large sieve
HTML articles powered by AMS MathViewer
- by Hugh L. Montgomery PDF
- Bull. Amer. Math. Soc. 84 (1978), 547-567
References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957 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)
- 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. (N.S.) 61 (103) (1963), 418–425 (Russian). MR 0171765
- M. B. Barban, The “large sieve” method and its application to number theory, Uspehi Mat. Nauk 21 (1966), no. 1, 51–102 (Russian). MR 0199171
- Richard Bellman, Almost orthogonal series, Bull. Amer. Math. Soc. 50 (1944), 517–519. MR 10639, DOI 10.1090/S0002-9904-1944-08180-9
- R. P. Boas Jr., A general moment problem, Amer. J. Math. 63 (1941), 361–370. MR 3848, DOI 10.2307/2371530
- E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225. MR 197425, DOI 10.1112/S0025579300005313
- Enrico Bombieri, Nuovi metodi e nuovi risultati nella teoria dei numeri, Boll. Un. Mat. Ital. (4) 1 (1968), 96–106 (Italian). MR 0234928
- Enrico Bombieri, On a theorem of van Lint and Richert, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 175–180. MR 0279065
- E. Bombieri, A note on the large sieve, Acta Arith. 18 (1971), 401–404. MR 286773, DOI 10.4064/aa-18-1-401-404
- E. Bombieri, On large sieve inequalities and their applications, Proceedings of the International Conference Number Theory (Moscow, 1971), 1973, pp. 251–256, 266 (English, with Russian summary). MR 0404176
- Enrico Bombieri, Le grand crible dans la théorie analytique des nombres, Astérisque, No. 18, Société Mathématique de France, Paris, 1974 (French). Avec une sommaire en anglais. MR 0371840
- E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. London Ser. A 293 (1966), 1–18. MR 199165, DOI 10.1098/rspa.1966.0155
- 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 0260703
- E. Bombieri and H. Davenport, Some inequalities involving trigonometrical polynomials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 23 (1969), 223–241. MR 249391
- D. A. Burgess, The average of the least primitive root modulo $p^{2}$, Acta Arith. 18 (1971), 263–271. MR 291118, DOI 10.4064/aa-18-1-263-271
- Chen Jing-run, 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
- Jing Run Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176. MR 434997
- Harold Davenport, Multiplicative number theory, Lectures in Advanced Mathematics, No. 1, Markham Publishing Co., Chicago, Ill., 1967. Lectures given at the University of Michigan, Winter Term, 1966. MR 0217022
- H. Davenport, The zeros of trigonometrical polynomials, Mathematika 19 (1972), 88–90. MR 316953, DOI 10.1112/S0025579300004988
- H. Davenport and H. Halberstam, The values of a trigonometrical polynomial at well spaced points, Mathematika 13 (1966), 91–96. MR 197427, DOI 10.1112/S0025579300004277
- H. Davenport and H. Halberstam, Primes in arithmetic progressions, Michigan Math. J. 13 (1966), 485–489. MR 200257, DOI 10.1307/mmj/1028999608
- P. D. T. A. Elliott, The Turan-Kubilius inequality, and a limitation theorem for the large sieve, Amer. J. Math. 92 (1970), 293–300. MR 263760, DOI 10.2307/2373324
- P. D. T. A. Elliott, On inequalities of large sieve type, Acta Arith. 18 (1971), 405–422. MR 286774, DOI 10.4064/aa-18-1-405-422
- P. D. T. A. Elliott, On connections between the Turán-Kubilius inequality and the large sieve: some applications, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 77–82. MR 0344215
- Pál Erdős, Remarks on number theory. V. Extremal problems in number theory. II, Mat. Lapok 17 (1966), 135–155 (Hungarian, with English summary). MR 217038
- P. Erdős and A. Rényi, Some remarks on the large sieve of Yu. V. Linnik, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 11 (1968), 3–13. MR 241378
- M. Forti and C. Viola, On large sieve type estimates for the Dirichlet series operator, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 31–49. MR 0506095
- P. X. Gallagher, The large sieve, Mathematika 14 (1967), 14–20. MR 214562, DOI 10.1112/S0025579300007968
- P. X. Gallagher, Bombieri’s mean value theorem, Mathematika 15 (1968), 1–6. MR 237442, DOI 10.1112/S002557930000231X
- P. X. Gallagher, A large sieve density estimate near $\sigma =1$, Invent. Math. 11 (1970), 329–339. MR 279049, DOI 10.1007/BF01403187
- P. X. Gallagher, Sieving by prime powers, Acta Arith. 24 (1973/74), 491–497. MR 337844, DOI 10.4064/aa-24-5-491-497
- P. X. Gallagher, The large sieve and probabilistic Galois theory, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 91–101. MR 0332694 34. S. W. Graham, Applications of sieve methods, Ph.D. Dissertation, Univ. of Michigan, Ann Arbor, 1977.
- H. Halberstam, The large sieve, Number Theory (Colloq., János Bolyai Math. Soc., Debrecen, 1968) North-Holland, Amsterdam, 1970, pp. 123–131. MR 0272744
- H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
- H. Halberstam and K. F. Roth, Sequences. Vol. I, Clarendon Press, Oxford, 1966. MR 0210679
- E. Hlawka, Bemerkungen zum großen Sieb von Linnik, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 178 (1970), 13–18 (German). MR 265314
- Edmund Hlawka, Zum großen Sieb von Linnik, Acta Arith. 27 (1975), 89–100 (German). MR 366845, DOI 10.4064/aa-27-1-89-100
- Christopher Hooley, On the Barban-Davenport-Halberstam theorem. I, J. Reine Angew. Math. 274(275) (1975), 206–223. MR 382202, DOI 10.1515/crll.1975.274-275.206
- M. N. Huxley, The large sieve inequality for algebraic number fields, Mathematika 15 (1968), 178–187. MR 237455, DOI 10.1112/S0025579300002540
- 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, DOI 10.1112/plms/s3-21.1.108
- 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, DOI 10.1112/jlms/s2-3.2.233
- M. N. Huxley, The distribution of prime numbers, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1972. Large sieves and zero-density theorems. MR 0444593
- M. N. Huxley, Irregularity in sifted sequences, J. Number Theory 4 (1972), 437–454. MR 311618, DOI 10.1016/0022-314X(72)90035-2
- A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), no. 1, 367–379. MR 1545625, DOI 10.1007/BF01180426
- John Johnsen, On the large sieve method in $\textrm {GF}[q,\,x]$, Mathematika 18 (1971), 172–184. MR 302617, DOI 10.1112/S0025579300005428
- I. Kobayashi, Remarks on the large sieve method, Seminar on Modern Methods in Number Theory (Inst. Statist. Math., Tokyo, 1971) Inst. Statist. Math., Tokyo, 1971, pp. 3. MR 0382206
- Isamu Kobayashi, A note on the Selberg sieve and the large sieve, Proc. Japan. Acad. 49 (1973), 1–5. MR 0325556
- I. P. Kubilyus, Probabilistic methods in the theory of numbers, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 2(68), 31–66 (Russian). MR 0079025
- U. V. Linnik, “The large sieve.”, C. R. (Doklady) Acad. Sci. URSS (N.S.) 30 (1941), 292–294. MR 0004266
- U. V. Linnik, A remark on the least quadratic non-residue, C. R. (Doklady) Acad. Sci. URSS (N.S.) 36 (1942), 119–120. MR 0007758
- J. Marcinkiewicz and A. Zygmund, Proof of a gap theorem, Duke Math. J. 4 (1938), no. 3, 469–472. MR 1546068, DOI 10.1215/S0012-7094-38-00439-9
- K. R. Matthews, On a bilinear form associated with the large sieve, J. London Math. Soc. (2) 5 (1972), 567–570. MR 318082, DOI 10.1112/jlms/s2-5.3.567
- K. R. Matthews, On an inequality of Davenport and Halberstam, J. London Math. Soc. (2) 4 (1972), 638–642. MR 302582, DOI 10.1112/jlms/s2-4.4.638
- K. R. Matthews, Hermitian forms and the large and small sieves, J. Number Theory 5 (1973), 16–23. MR 321895, DOI 10.1016/0022-314X(73)90054-1
- Ming-chit Liu, On a result of Davenport and Halberstam, J. Number Theory 1 (1969), 385–389. MR 249392, DOI 10.1016/0022-314X(69)90001-8
- H. L. Montgomery, A note on the large sieve, J. London Math. Soc. 43 (1968), 93–98. MR 224585, DOI 10.1112/jlms/s1-43.1.93
- H. L. Montgomery, Mean and large values of Dirichlet polynomials, Invent. Math. 8 (1969), 334–345. MR 268130, DOI 10.1007/BF01404637
- H. L. Montgomery, Zeros of $L$-functions, Invent. Math. 8 (1969), 346–354. MR 249375, DOI 10.1007/BF01404638
- Hugh L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR 0337847, DOI 10.1007/BFb0060851
- H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134. MR 374060, DOI 10.1112/S0025579300004708
- H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–82. MR 337775, DOI 10.1112/jlms/s2-8.1.73
- H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach’s problem, Acta Arith. 27 (1975), 353–370. MR 374063, DOI 10.4064/aa-27-1-353-370
- Yoichi Motohashi, A note on the large sieve, Proc. Japan Acad. 53 (1977), no. 1, 17–19. MR 432571
- Yoichi Motohashi, On Gallagher’s prime number theorem, Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), no. 2, 50–52. MR 480383
- Yoichi Motohashi, Introduction to the theory of the distribution of prime numbers, Sūgaku 26 (1974), no. 1, 1–12 (Japanese). MR 414501
- Yoichi Motohashi, On a density theorem of Linnik, Proc. Japan. Acad. 51 (1975 suppl), 815–817. MR 0401677, DOI 10.2183/pjab1945.51.Supplemnt_{8}15
- Yoichi Motohashi, A note on the large sieve. II, Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), no. 4, 122–124. MR 485749
- 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, DOI 10.1090/coll/019
- P. A. B. Pleasants, A sum related to the distribution modulo $1$ of sets of real numbers, Quart. J. Math. Oxford Ser. (2) 21 (1970), 321–336. MR 271062, DOI 10.1093/qmath/21.3.321
- A. A. Ren′i, On the representation of an even number as the sum of a single prime and a single almost-prime number, Doklady Akad. Nauk SSSR (N.S.) 56 (1947), 455–458 (Russian). MR 0021958
- A. Rényi, On the representation of an even number as the sum of a prime and of an almost prime, Amer. Math. Soc. Transl. (2) 19 (1962), 299–321. MR 0131413, DOI 10.1090/trans2/019/12
- Alfred Rényi, Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres, J. Math. Pures Appl. (9) 28 (1949), 137–149 (French). MR 31505
- Alfréd Rényi, Probability methods in number theory, Publ. Math. Collectae Budapest 1 (1949), no. 21, 9. MR 0036782
- Alfred Rényi, On a theorem of the theory of probability and its application in number theory, Časopis Pěst. Mat. Fys. 74 (1949), 167–175 (1950) (Russian, with Czech summary). MR 0039750
- Alfred Rényi, Sur un théorème général de probabilité, Ann. Inst. Fourier (Grenoble) 1 (1949), 43–52 (1950) (French). MR 54187, DOI 10.5802/aif.6
- Alfred Rényi, On the large sieve of Ju V. Linnik, Compositio Math. 8 (1950), 68–75. MR 34407
- Alfréd Rényi, On the probabilistic generalization of the large sieve of Linnik, Magyar Tud. Akad. Mat. Kutató Int. Közl. 3 (1958), 199–206 (English, with Russian and Hungarian summaries). MR 111073
- A. Rényi, Probabilistic methods in number theory, Proc. Internat. Congress Math. 1958., Cambridge Univ. Press, New York, 1960, pp. 529–539. MR 0118707
- A. Rényi, New version of the probabilistic generalization of the large sieve, Acta Math. Acad. Sci. Hungar. 10 (1959), 217–226 (English, with Russian summary). MR 111074, DOI 10.1007/BF02063300
- G. J. Rieger, Zum Sieb von Linnik, Arch. Math. (Basel) 11 (1960), 14–22 (German). MR 153632, DOI 10.1007/BF01236901
- G. J. Rieger, Das grosse Sieb von Linnik für algebraische Zahlen, Arch. Math. (Basel) 12 (1961), 184–187 (German). MR 133318, DOI 10.1007/BF01650547
- P. M. Ross, On Chen’s theorem that each large even number has the form $p_{1}+p_{2}$ or $p_{1}+p_{2}p_{3}$, J. London Math. Soc. (2) 10 (1975), no. 4, 500–506. MR 389816, DOI 10.1112/jlms/s2-10.4.500
- K. F. Roth, Remark concerning integer sequences, Acta Arith. 9 (1964), 257–260. MR 168545, DOI 10.4064/aa-9-3-257-260
- K. F. Roth, On the large sieves of Linnik and Rényi, Mathematika 12 (1965), 1–9. MR 197424, DOI 10.1112/S0025579300005088
- K. F. Roth, The large sieve, Imperial College of Science and Technology, London, 1968. Inaugural Lecture, 23 January, 1968. MR 0237458
- A. G. Samandarov, The large sieve in algebraic number fields, Mat. Zametki 2 (1967), 673–680 (Russian). MR 223331
- Werner Schaal, On the large sieve method in algebraic number fields, J. Number Theory 2 (1970), 249–270. MR 272745, DOI 10.1016/0022-314X(70)90052-1 90. I. Schur, Bemerkungen zur Theorie der beschrankten Bilinearformen mit unendlich vielen Verändlichen, J. Reine Angew. Math. 140 (1911), 1-28.
- Wolfgang Schwarz, Einführung in Siebmethoden der analytischen Zahlentheorie, Bibliographisches Institut, Mannheim-Vienna-Zurich, 1974. MR 0409392
- S. L. Sobolev, Applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by F. E. Browder. MR 0165337, DOI 10.1090/mmono/007
- A. V. Sokolovskiĭ, The “large sieve”, Acta Arith. 25 (1973/74), 301–306 (Russian). MR 347755 94. E. C. Titchmarsh, A class of trigonometrical series, J. London Math. Soc. 3 (1928), 300-304.
- Saburô Uchiyama, The maximal large sieve, Hokkaido Math. J. 1 (1972), 117–126. MR 321896, DOI 10.14492/hokmj/1535508578 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.
- Norbert Wiener, A class of gap theorems, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 3 (1934), no. 3-4, 367–372. MR 1556735
- Robin J. Wilson, The large sieve in algebraic number fields, Mathematika 16 (1969), 189–204. MR 263774, DOI 10.1112/S0025579300008160
- Dieter Wolke, Farey-Brüche mit primem Nenner und das große Sieb, Math. Z. 114 (1970), 145–158 (German). MR 260704, DOI 10.1007/BF01110322
- Dieter Wolke, Einige Anwendungen des großen Siebes auf zahlentheoretische Funktionen, Philipps-Universität, Marburg/Lahn, 1970 (German). Habilitationsschrift. MR 0279058
- D. Wolke, On the large sieve with primes, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239–247. MR 291121, DOI 10.1007/BF01896016
- Dieter Wolke, Über eine Ungleichung von A. I. Vinogradov, Arch. Math. (Basel) 23 (1972), 625–629 (German). MR 321886, DOI 10.1007/BF01304943
- Dieter Wolke, A lower bound for the large sieve inequality, Bull. London Math. Soc. 6 (1974), 315–318. MR 354587, DOI 10.1112/blms/6.3.315
Additional Information
- 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