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The large sieve inequality is equivalent to the bound for the largest eigenvalue of the matrix , naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is . Based on his numerical data Ramaré conjectured that when as for some finite positive constant , the limiting distribution of the eigenvalues of , scaled by , exists and is non-degenerate. In this paper we prove this conjecture by establishing the convergence of all moments of the eigenvalues of as . 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 . Some of the main ingredients in our proof include the large-sieve inequality and results on -correlations of Farey fractions.
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Florin P. Boca, Maksym Radziwiłł, Limiting distribution of eigenvalues in the large sieve matrix. J. Eur. Math. Soc. 22 (2020), no. 7, pp. 2287–2329DOI 10.4171/JEMS/965