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Limiting distribution of eigenvalues in the large sieve matrix

Boca, Florin P. and Radziwiłł, Maksym (2016) Limiting distribution of eigenvalues in the large sieve matrix. . (Submitted) https://resolver.caltech.edu/CaltechAUTHORS:20180612-154327663

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Abstract

The large sieve inequality is equivalent to the bound λ_1 ⩽ N+Q^2−1 for the largest eigenvalue λ_1 of the N by 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≍Q^2. Based on his numerical data Ramaré conjectured that when N ∼ αQ^2 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.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/1609.05843arXivDiscussion Paper
Additional Information:The authors are grateful to Alexandru Zaharescu for stimulating discussions at the beginning of this project and for making this collaboration possible. The work of the first author was supported in part by the CNCS-UEFISCDI project PN-IIID-PCE-2012-4-0201 and by a one month Bitdefender Invited Professor Scholarship held at IMAR Bucharest.
Funders:
Funding AgencyGrant Number
Consiliul Național al Cercetării Științifice (CNSC)PN-IIID-PCE-2012-4-0201
Institute of Mathematics of the Romanian Academy (IMAR)UNSPECIFIED
Record Number:CaltechAUTHORS:20180612-154327663
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20180612-154327663
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87035
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Jun 2018 22:56
Last Modified:03 Oct 2019 19:51

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