Limiting distribution of eigenvalues in the large sieve matrix

Florin P. Boca, Maksym Radziwiłł

Research output: Contribution to journalArticlepeer-review

Abstract

The large sieve inequality is equivalent to the bound λ1 ≤ N + Q2 − 1 for the largest eigenvalue λ1 of the N × 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 Q2. Based on his numerical data, Ramaré conjectured that when N ∼ αQ2 as Q → ∞ for some finite positive constant α, 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 → ∞. 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. Some of the main ingredients in our proof include the large-sieve inequality and results on n-correlations of Farey fractions.

Original languageEnglish (US)
Pages (from-to)2287-2329
Number of pages43
JournalJournal of the European Mathematical Society
Volume22
Issue number7
DOIs
StatePublished - 2020

Keywords

  • Correlations
  • Eigenvalues distribution
  • Farey fractions
  • Large sieve matrix

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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