Limiting distribution of eigenvalues in the large sieve matrix

TL;DR

This paper proves the conjecture that the eigenvalues of a large sieve matrix, scaled appropriately, have a well-defined limiting distribution as the matrix size grows, with moments explicitly characterized.

Contribution

It establishes the convergence of all eigenvalue moments of the large sieve matrix and describes the limiting distribution explicitly, confirming Ramaré's conjecture.

Findings

Eigenvalues scaled by 1/N have a non-degenerate limiting distribution.

All moments of the eigenvalues converge as Q approaches infinity.

The moments vary continuously with the parameter α.

Abstract

The large sieve inequality is equivalent to the bound $\lambda_1 \leqslant N + Q^2-1$ for the largest eigenvalue $\lambda_1$ of the $N$ by $N$ matrix $A^{\star} A$, 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 Ramar\'e conjectured that when $N \sim \alpha Q^2$ as $Q \rightarrow \infty$ for some finite positive constant $\alpha$, the limiting distribution of the eigenvalues of $A^{\star} A$, 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 $A^{\star} A$ as $Q\rightarrow\infty$. Previously only the second moment was known, due to Ramar\'e. 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.