The large sieve

Notes on the large sieve

Large optical photon sieve.

A photon sieve with 10(7) holes has been constructed for operation at optical wavelengths. Details of the design, fabrication, and performance of this device are presented. The 1 m focal-length, 0.1 m diameter element is diffraction limited over a significant bandwidth and has a moderate field of view.

The Principle of the Large Sieve

We describe a very general abstract form of sieve based on a large-sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We give some first results on the number of prime divisors of “most” elements of an elliptic divisib...

Large Deviations in the Random Sieve

The proportion k of gaps with length k between square-free numbers is shown to satisfy log k = 1 + o(1) (6== 2)k log k as k ! 1. Such asymptotics are consistent with Erd} os's challenge to prove that the gap following the square-free number t is smaller than c log t= log log t, for all t and some constant c satisfying c > 2 =12. The results of this paper are achieved by studying the probabiliti...

Eigenvalues in the large sieve inequality, II

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