JournalsjemsVol. 22, No. 7pp. 2287–2329

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
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The large sieve inequality is equivalent to the bound λ1N+Q21\lambda_1 \leq N + Q^2-1 for the largest eigenvalue λ1\lambda_1 of the N×NN \times N matrix AAA^*A, naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is NQ2N \asymp Q^2. Based on his numerical data Ramaré conjectured that when NαQ2N \sim \alpha Q^2 as QQ \to \infty for some finite positive constant α\alpha, the limiting distribution of the eigenvalues of AAA^*A, scaled by 1/N1/N, exists and is non-degenerate. In this paper we prove this conjecture by establishing the convergence of all moments of the eigenvalues of AAA^*A as QQ\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 α\alpha. Some of the main ingredients in our proof include the large-sieve inequality and results on nn-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