# Limiting distribution of eigenvalues in the large sieve matrix

### Florin P. Boca

University of Illinois at Urbana-Champaign, USA### Maksym Radziwiłł

McGill University, Montreal, Canada

## Abstract

The large sieve inequality is equivalent to the bound $λ_{1}≤N+Q_{2}−1$ for the largest eigenvalue $λ_{1}$ of the $N×N$ matrix $A_{∗}A$, naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is $N≍Q_{2}$. Based on his numerical data Ramaré conjectured that when $N∼αQ_{2}$ as $Q→∞$ for some finite positive constant $α$, 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 $Q→∞$. 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 $α$. Some of the main ingredients in our proof include the large-sieve inequality and results on $n$-correlations of Farey fractions.

## Cite this article

Florin P. Boca, Maksym Radziwiłł, Limiting distribution of eigenvalues in the large sieve matrix. J. Eur. Math. Soc. 22 (2020), no. 7, pp. 2287–2329

DOI 10.4171/JEMS/965