Limiting distribution of eigenvalues in the large sieve matrix

Abstract

The large sieve inequality is equivalent to the bound \lambda_1 \leq N + Q^2-1 for the largest eigenvalue \lambda_1 of the N \times N matrix A^*A , naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is N \asymp Q^2 . Based on his numerical data Ramaré conjectured that when N \sim \alpha Q^2 as Q \to \infty for some finite positive constant \alpha , the limiting distribution of the eigenvalues of A^*A , scaled by 1/N , exists and is non-degenerate. In this paper we prove this conjecture by establishing the convergence of all moments of the eigenvalues of A^*A as Q\to\infty . Previously only the second moment was known, due to Ramaré. Furthermore, we obtain an explicit description of the moments of the limiting distribution, and establish that they vary continuously with \alpha . Some of the main ingredients in our proof include the large-sieve inequality and results on n -correlations of Farey fractions.