## The sharp large sieve inquality

This is the first post in a short sequence related to the large sieve inequality and its applications. In this post I will give a proof of the sharp (analytic) large sieve inequality. While the main result of this post is a purely analytic statement, one can quickly obtain a large number of arithmetic consequences from it. The most famous of these is probably a theorem of Brun which states that the sum of the reciprocals of the twin primes converges. I will present this and other arithmetic applications in a following post. This post will focus on proving the sharp (analytic) large sieve inequality. Much of the work in this post could be simplified if we were willing to settle for a less than optimal constant in the inequality. This would have little impact on the the arithmetic applications, but we’ll stick to proving the sharp form of this inequality (mostly for my own benefit). I should point out that there is an alternate approach to this inequality, via Selberg-Buerling extremal functions, which gives an even sharper result. In particular, using this method one can replace the in Theorem 1 below by . We will not pursue this further here, however. Much of this post will follow this paper of Montgomery, however I have made a few attempts at simplifying the presentation, most notably I have eliminated the need to use properties of skew-hermitian forms in the proof of the sharp discrete Hilbert transform inequality.

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