The analytic principle of the large sieve

Abstract

E. Bombieri [12] has written at length concerning applications of the large sieve to number theory. Our intent here is to complement his exposition by devoting our attention to the analytic principle of the large sieve; we describe only briefly how applications to number theory are made. The large sieve was studied intensively during the decade [1965][1966][1967][1968][1969][1970][1971][1972][1973][1974][1975], with the result that the subject has lost its mystery: We now possess a variety of simple ideas which provide very precise results and a host of variants. While the large sieve can no longer be considered deep, it nevertheless gives powerful estimates in many different settings. ) has obtained k * 2. In all of these arguments the large sieve is a major tool.) The large sieve remained the province of a few specialists, until the appearance in 1965 of a fundamental paper of K. F. Roth [86], followed immediately by a major contribution of Bombieri [7]. As we consider it here, the large sieve was first reduced to its basic analytic principle by H. Davenport and H. Halberstam [21].2. The nature of the large sieve. For M + 1 < n < M + AT we let a n be arbitrary complex numbers, and we form the trigonometric polynomial M+N S (a) -2 a n e(na);here e(9) = e 2 ™ e , so that S (a) has period 1. Let a" ..., a R be points which are well spaced (mod 1) in the sense that This is an expanded account of an invited address entitled The large sieve for the mathematician in the street,

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