Limiting distribution of eigenvalues in the large sieve matrix

Abstract

The large sieve inequality is equivalent to the bound ${\lambda }_1\le N+Q^2-1$ for the largest eigenvalue ${\lambda }_1$ of the $N\times N$ matrix $A^{\ast }A$, 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 $A^{\ast }A$, 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 $A^{\ast }A$ 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.