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Journal of the European Mathematical Society

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Volume 22, Issue 7, 2020, pp. 2287–2329
DOI: 10.4171/JEMS/965

Published online: 2020-04-15

Limiting distribution of eigenvalues in the large sieve matrix

Florin P. Boca[1] and Maksym Radziwiłł[2]

(1) University of Illinois at Urbana-Champaign, USA
(2) McGill University, Montreal, Canada

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.

Keywords: Large sieve matrix, eigenvalues distribution, Farey fractions, correlations

Boca Florin, Radziwiłł Maksym: Limiting distribution of eigenvalues in the large sieve matrix. J. Eur. Math. Soc. 22 (2020), 2287-2329. doi: 10.4171/JEMS/965