Eigenvalues in the large sieve inequality

Abstract

We provide some evidence that the eigenvalues of the hermitian form $\sum_{a/q}|\sum_{n\le N}\varphi_ne(na/q)|^2$ tend to have a limit distribution when $N$ and $Q$ go simultaneously to infinity in such a way that $N/Q^2$ tends to a constant. We also present some background material, as well as a large sieve equality, when $N\Log^7 N = o(Q)$, that follows from our results.