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These are lecture notes from a graduate course in sieve methods which I delivered in the spring of 1996 at Rutgers. Though generally regarded as part of elementary number theory the sieve methods are not quite as easy to teach because of complexity of arguments and diversity of techniques. A modern graduate student has little, if any, exposure to this nevertheless fascinating area. Thus I assumed nothing of the student at the start. For five out of nine chapters I provided appendices in which a vast side material is presented that is needed to make the exposition of sieve theory self-contained to a great extent. Anyone who studies these notes from scratch should be able to check proofs thoroughly, even though some technical details are being suppressed gradually toward the end.
The sieve theory as well as its applications are equally interesting. Of course, one cannot practice the latter without the former. Therefore we spend considerable time in developing theoretical background. In fact I have placed strong emphasis to generality, however, not as far as to lose touch with practice. To make the theory intelligible I introduce the involved concepts in an intuitive fashion rather than by abstract definitions this approach serves the purpose without compromising the clarity and precision of results needed for applications