Lower Bound Sieves when κ > 1

Abstract

The methods of this chapter stem from the observation, derived in part from numerical evidence, that, when the sifting density κ exceeds 1 and for the smaller values of s, Selberg’s upper bound for S(A, P(D ^1/s)) is better than that derived by Rosser’s method. We therefore seek to apply Selberg’s ideas also to the lower bound sifting problem. There is more than one way in which this can be attempted, of which we will introduce two. In neither of these cases do we attempt a complete account of all the work to be found in the existing literature, partly for reasons of space and partly because it seems not to be clear that the best approach to the question has yet been found.