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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 formarising in the inequality. For arithmetic applications the most interestingrange is $N \asymp Q^2$. Based on his numerical data Ramar\'e conjectured thatwhen $N \sim \alpha Q^2$ as $Q \rightarrow \infty$ for some finite positiveconstant $\alpha$, 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 allmoments of the eigenvalues of $A^{\star} A$ as $Q\rightarrow\infty$. Previouslyonly the second moment was known, due to Ramar\'e. Furthermore, we obtain anexplicit description of the moments of the limiting distribution, and establishthat they vary continuously with $\alpha$. Some of the main ingredients in ourproof include the large-sieve inequality and results on $n$-correlations ofFarey fractions.

Limiting distribution of eigenvalues in the large sieve matrix