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The large sieve in analytic number theory. (Le grand crible dans la théorie analytique des nombres.) (French) Zbl 0292.10035
Astérisque 18, 1-87 (1974).
In this monograph the author presents a concise and beautiful account of the Large Sieve and some of its important applications to analytic number theory, in particular, the density theorems, the distribution of primes in arithmetical progressions and the connection with the small sieve of Brun-Selberg. The method of the Large Sieve, which was invented by Yu. V. Linnik [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 30, 292–294 (1941; Zbl 0024.29302)] and has later been developed by A. Renyi, M. B. Barban, A. I. Vinogradov, K. F. Roth, E. Bombieri, P. X. Gallagher and others, is, as is widely recognized, one of the most powerful tools in multiplicative number theory. The booklet consists of Introduction, Notes in twelve paragraphs, Bibliography and Summary in English. The twelve paragraphs are: §0. Preliminaires. Notations and a definition of a sieve; §1. Quelques exemples. Le crible de Linnik et Renyi. The theorem of Linnik an the least quadratic non-residue (mod \(p\)), and Renyi’s formulation of the large sieve; §2. La forme analytique additive du grand crible. A formulation of the large sieve due to Roth and Bombieri; §3. Applications arithmetiques. Le crible de Selberg (I); §4. La forme multiplicative du grand crible. The large sieve inequality containing Dirichlet’s residue characters; §5. La forme analytique multiplicative du grand crible. The so-called hybrid sieve of Gallagher and some results due to M. Forti and C. Viola; §6. Applications. Le theoreme de Linnik. A density theorem for zeros of Dirichlet \(L\)-functions and the famous theorem of Linnik an the least prime in an arithmetic progression; §7. Applications. Le théorème des nombres premiers dans les progressions arithmétiques. A modification of the simplified proof by Gallagher of the Bombieri-Vinogradov theorem; §8. Le crible de Selberg (II). A generalization of the classical sieve of Selberg; §9. Application du crible de Selberg. A proof of the result that there are infinitely many primes \(p\) such that \(p + 2\) has at most \(4\) prime factors; §10. Théorèmes de densité; §11. Notes Bibliographiques. The results demonstrated are not always the strongest ones known at present, but the exposition is an the whole very clear and readable.
Reviewer: Saburo Uchiyama

MSC:
11N35 Sieves
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11M35 Hurwitz and Lerch zeta functions
11N13 Primes in congruence classes
11P32 Goldbach-type theorems; other additive questions involving primes