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On the large sieves of Linnik and Rényi

Published online by Cambridge University Press:  26 February 2010

K. F. Roth
Affiliation:
University College, London

Extract

Let N be a natural number, and let be a non-empty subset of the set of integers

Let Z be the number of elements of . We denote by Z(p, h) the number of elements of falling into the congruence class h modulo p, so that

Type
Research Article
Copyright
Copyright © University College London 1965

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References

1. JuLimiik, V., “The large sieve”, Cmnptes Rendus (Doklady) de l'Academie des Sciences de l'URSS, 30(1941), 292294.Google Scholar
2. Rényi, A, “On the representation of even numbers as the sum of a prime and an almost prime number”, Bulletin (Izvestia) de l'URSS, Ser. Math., 12 (1948), 5758. (In Russian, but an Amer. Math. Soc. translation is available: Selected transl. ser. II vol. 19).Google Scholar
3. Rényi, A, “On the large sieve of Ju. V. Linnik”, Compositio Math., 8 (1950), 6875.Google Scholar
4. Rényi, A, “New version of the probabilistic generalization of the large sieve”, Acta Math. Acad. Sei. Hungar., 10 (1959), 217226. (This paper contains a comprehensive list of references, and in particular, references for the earlier versions of Rényi's probabilistic approach.)CrossRefGoogle Scholar
5. Rényi, A, “Probabilistic methods in number theory”, Progress in Math., 4 (1958), 465510. (In Chinese.)Google Scholar
6. Halberstam, H. and Roth, K. F., “Sequences”, Clarondon Press Oxford (in the course of publication) Chapter IV, §10.Google Scholar
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