Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T15:57:30.786Z Has data issue: false hasContentIssue false
  • English
  • Français

The large sieve

Published online by Cambridge University Press:  26 February 2010

H. L. Montgomery  and
R. C. Vaughan
H. L. Montgomery
Affiliation:
The University of Michigan, Ann Arbor, Michigan, U.S.A.
R. C. Vaughan
Affiliation:
Imperial College, London
Get access

Extract

Let S(x) be a trigonometric polynomial,

where N > 0 and M are integers, the an are arbitrary complex numbers, and e(x) = e2πix. In its basic form, the large sieve of Linnik and Rényi is an inequality of the form

Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 119 - 134
Copyright
Copyright © University College London 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bombieri, E., “A note on the large sieve ”, Acta Arith., 18 (1971), 401–404.CrossRefGoogle Scholar
2. Bombieri, E. and Davenport, H., “On the large sieve method ”, Abh. aus Zahlentheorie und Analysis Zur Erinnerung an Edmund Landau, Deut. Verlag Wiss., Berlin, 1968, 11–22.CrossRefGoogle Scholar
3. Bombieri, E. and Davenport, H., “Some inequalities involving trigonometrical polynomials”, Ann. Scuola Norm. Sup. Pisa, 23 (1969), 223–241.Google Scholar
4. Davenport, H., Multiplicative number theory (Markham, Chicago, 1967).Google Scholar
5. Elliott, P. D. T. A., “Some remarks concerning the large sieve type”, Acta Arith., 18 (1971), 405–422.CrossRefGoogle Scholar
6. Gallagher, P. X., “The large sieve”, Mathematika, 14 (1967), 14–20.CrossRefGoogle Scholar
7. Hardy, G. H.Littlewood, J. E. and Pólya, G., Inequalities (University Press, Cambridge, 1934).Google Scholar
8. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, Fourth edition (Clarendon Press, Oxford, 1964).Google Scholar
9. Hellinger, E. and Toeplitz, O., “Grundlagen fur eine Theorie der unendlichen Matrizen”, Math. Ann., 69 (1910), 289–330.CrossRefGoogle Scholar
10. Hensley, D. and Richards, I., “On the incompatibility of two conjectures concerning primes”, Proc Symposia Pure Math., 24 (1973), 123–128.CrossRefGoogle Scholar
11. Klimov, N. I., “Almost prime numbers”, Uspehi Mat. Nauk, 16 (3), (99) (1961), 181–188. See Amer. Math. Soc Transl., (92) 46 (1965), 48–56.Google Scholar
12. Kobayashi, I., “Remarks on the large sieve method”, Proc. Japan-United States Seminar on Number Theory, 1971, 3 pp.Google Scholar
13. Lint, J. H. van and Richert, H.-E., “On primes in arithmetic progressions”, Acta Arith., 11 (1965), 209–216.CrossRefGoogle Scholar
14. Matthews, K. R., “On an inequality of Davenport and Halberstam”, J. London Math. Soc., (2) 4 (1972), 638–642.CrossRefGoogle Scholar
15. Matthews, K. R., “On a bilinear form associated with the large sieve”, London Math. Soc., (2) 5 (1972), 567–570.CrossRefGoogle Scholar
16. Matthews, K. R., “Hermitian forms and the large and small sieves”, J. Number Theory, 5 (1973), 16–23.CrossRefGoogle Scholar
17. Liu, Ming-Chit, “On a result of Davenport and Halberstam”, Number Theory, 1 (1969), 385–389.CrossRefGoogle Scholar
18. Montgomery, H. L., “A note on the large sieve” J. London Math. Soc., 43 (1968), 93–98.CrossRefGoogle Scholar
19. Montgomery, H. L., Topics in multiplicative number theory, Lecture Notes in Mathematics, 227 (Springer- Verlag, 1971).CrossRefGoogle Scholar
20. Montgomery, H. L. and Vaughan, R. C., “Hilbert's inequality ”, J. London Math. Soc. (to appear).Google Scholar
21. Rosser, J. B. and Schoenfeld, L., “Approximate formulas for some functions of prime numbers”, Illinois J. Math., 6 (1962), 64–94.CrossRefGoogle Scholar
22. Schinzel, A., “Remarks on the paper ‘Sur certaines hypothèles concernant les nombres premiers ’”, Acta Arith., 7 (1961), 1–8.CrossRefGoogle Scholar
23. Schinzel, A.Sierpiński, and W., “Sur certaines hypotheses concernant les nombres premiers”, Acta Arith., 4 (1958), 185–208.CrossRefGoogle Scholar
24. Selberg, A., “On elementary methods in prime number theory and their limitations ”, Den 11-te Skand. Mat.-Kongress, Trondhjem, 1949, 13–22.Google Scholar
25. Selberg, A., “The general sieve-method and its place in prime number theory”, Proc. Intern. Congr.Math., Cambridge, Mass., 1950, 1, 286–292.Google Scholar
26. Uchiyama, S., “A note on the sieve method of A. Selberg”, J. Fac. Sci. Hokkaido Univ., (1) 16 (1962), 189–192.Google Scholar
27. Ward, D. R., “Some series involving Euler's function”, J. London Math. Soc., 2 (1927) 210–214.CrossRefGoogle Scholar