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A comprehensive and up-to-date treatment of sieve methods. The theory of the sieve is developed thoroughly with complete and accessible proofs of the basic theorems. A wide range of applications are included, both to traditional questions, and to areas previously unexplored by sieve methods. New proofs are given of the central theorems of analytic number theory, these proofs emphasize and take advantage of the applicability of sieve ideas. No analytic number theorist should be without this volume.