Limiting distribution of eigenvalues in the large sieve matrix

The large sieve inequality is equivalent to the bound $\lambda_1 \leqslant N

Q^2-1 $f o r t h e l a r g e s t e i g e n v a l u e$ \lambda_1 $o f t h e$ N $b y$ N $m a t r i x$ A^{\star} 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 Ramar'e conjectured that when $N \sim α Q^{2}$ as $Q \to \infty$ for some finite positive constant $α$ , the limiting distribution of the eigenvalues of $A^{\star} A $, s c a l e d b y$ 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 Ramar'e. 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.

The large sieve inequality is equivalent to the bound $\lambda_1 \leqslant N

Q^2-1 $f o r t h e l a r g e s t e i g e n v a l u e$ \lambda_1 $o f t h e$ N $b y$ N $m a t r i x$ A^{\star} 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 Ramar'e conjectured that when $N \sim α Q^{2}$ as $Q \to \infty$ for some finite positive constant $α$ , the limiting distribution of the eigenvalues of $A^{\star} A $, s c a l e d b y$ 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 Ramar'e. 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.

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