Fractional power large sieves

Abstract

In the form given it by Davenport and Halberstam, [2], the inequality of the Large Sieve asserts that for points δ _j , j = 1,2,…, spaced according to ∥δ _j − δ_k∥ ≥ δ > 0, j ≠ k, the function

$$s(\alpha ) = \sum\limits_{n = - N}^N {a_n e^{2\pi in\alpha } ,} \,\,\alpha \,\text{real}$$

satisfies

$$\sum\nolimits_j {\left| {S(\alpha _j )} \right|^2 \leqslant 2.2\max (\delta ^{ - 1} ,2N)\,\sum\limits_{n = - N}^N {\left| {a_n } \right|^2 } }$$

((1))

Here N denotes a positive integer, ∥y∥ the distance of the real y from a nearest integer.