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The large sieve

Published online by Cambridge University Press:  26 February 2010

P. X. Gallagher
P. X. Gallagher
Affiliation:
Barnard College, Columbia University, New York, 27, New York, U.S.A.
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Extract

1. The purpose of this paper is to give simple proofs for some recent versions of Linnik's large sieve, and some applications.

The first theme of the large sieve is that an arbitrary set of Z integers in an interval of length N must be well distributed among most of the residue classes modulo p, for most small primes p, unless Z is small compared with N. Following improvements on Linnik's original result [1] by Rényi [2] and by Roth [3], Bombieri [4] recently proved the following inequality: Denote by Z(a, p) the number of integers in the set which are congruent to a modulo p.

Type
Research Article
Information
Mathematika , Volume 14 , Issue 1 , June 1967 , pp. 14 - 20
Copyright
Copyright © University College London 1967

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References

1.Linnik, Yu. V., “The large sieve”, Doklady Akad. Nauk SSSR, 30 (1941), 292–294.Google Scholar
2.Renyi, A., “On the large sieve of Ju. V. Linnik”, Compositio Math., 8 (1950), 68–75.Google Scholar
3.Roth, K. F., “On the large sieve of Linnik and Renyi”, Mathematika, 12 (1965), 1–9.CrossRefGoogle Scholar
4.Bombieri, E., “On the large sieve”, Mathematika, 12 (1965), 201–225.CrossRefGoogle Scholar
5.Lang, S. and Tate, J. T., The collected papers of Emil Artin (Addison-Wesley, 1965), Preface.Google Scholar
6.Davenport, H. and Halberstam, H., “The values of a trigonometric polynomial at well spaced points”, Mathematika, 13 (1966), 91–96.CrossRefGoogle Scholar
7.Davenport, H. and Halberstam, H., “Primes in arithmetic progressions”, Michigan Math. J., (to appear).Google Scholar
8.Paley, R. E. A. C., “On the analogues of some theorems in the theory of the Riemann ς-function”, Proc. Lond. Math. Soc, (2), 31 (1931), 273–311.CrossRefGoogle Scholar
9.Prachar, K., Primzahlverteilung (Springer, 1957).Google Scholar