On variants of the larger sieve

The large sieve has its origins in the work of Linnik and Rényi. It was developed to deal with sequences that avoid a positive proportion of residue classes. It was later simplified by Roth, Bombieri, Davenport, Halberstam, Montgomery, Gallagher and many others. For a survey see Montgomery [12] and Bombieri [1]. It is known that Montgomery’s large sieve [12] is a useful method when sifting sequences that avoid many residue classes modulo primes. For example, if a sequence A ⊂ [1, N ] avoids ω(p) = p−1 2 residue classes modulo the primes p ≤ √ N , then the large sieve gives the upper bound A(N) √ N . Moreover the squares are the standard example to show that here the large sieve achieves the correct order of magnitude. But for sequences that avoid on average more than half of the residue classes it is preferable to use Gallagher’s larger sieve [7]. Recently there emerged quite a few new applications of Gallagher’s larger sieve so that it seemed worthwhile looking for variants of this sieve having some advantages over Gallagher’s version. (We would like to point out that Gallagher had various contributions to the large sieve. In addition to the larger sieve that we use here, he gave a simplified version of the large sieve [6] and developed a sieve that allows to sift modulo powers of primes, [8].) Let us state Montgomery’s large sieve and Gallagher’s larger sieve first. We then state and prove our new variants of it. In the final section we discuss the advantages or disadvantages of these variants.