Sieve Methods

H. Halberstam and K. F. Roth
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H. Halberstam: University of Illinois, Department of Mathematics
K. F. Roth: Imperial College of Science and Technology, Department of Mathematics

Chapter IV in Sequences, 1983, pp 186-237 from Springer

Abstract: Abstract In this chapter we discuss the sieve methods of Viggo Brun and A. Selberg, and the ‘large sieves’ of Linnik and Rényi. Of these, the theorems of Linnik and Rényi in § 10 fall most naturally within the scope of this book; they are theorems of surprising generality, and are of intrinsic interest quite apart from their applications to sieve problems. The sieves of Brun and Selberg, on the other hand, are effective only when applied to sequences of rather a special kind. Nevertheless, the method is one of some generality and great beauty, and we therefore give an account of its mechanism. We do not include any applications to specific sieve problems, although we prove two general theorems of Selberg (Theorems 3 and 4) which are applicable to a wide variety of such problems. For applications in other chapters, we require only Theorem 1. Whilst this result is usually ascribed to the Brun-type sieve (and indeed can be derived in this way), we shall see that it can be proved by simpler and more direct means.

Keywords: Natural Number; Prime Factor; Lower Estimate; Congruence Class; Implied Constant (search for similar items in EconPapers)
Date: 1983
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DOI: 10.1007/978-1-4613-8227-0_4

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