The "visible" Wigner matrix

We consider the ``visible'' Wigner matrix, a Wigner matrix whose \((i, j)\)-th entry is coerced to zero if \(i, j\) are co-prime. Using a recent result from elementary number theory on co-primality patterns in integers, we show that the limiting spectral distribution of this matrix exists,...

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Bibliographic Details
Published in:arXiv.org 2023-12
Main Authors: Bose, Arup, Soumendu Sundar Mukherjee
Format: Article
Language:English
Subjects:
Number theory
Polynomials
Online Access:Get full text
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Summary:We consider the ``visible'' Wigner matrix, a Wigner matrix whose \((i, j)\)-th entry is coerced to zero if \(i, j\) are co-prime. Using a recent result from elementary number theory on co-primality patterns in integers, we show that the limiting spectral distribution of this matrix exists, and give explicit descriptions of its moments in terms of infinite products over primes \(p\) of certain polynomials evaluated at \(1/p\). We also consider the complementary ``invisible'' Wigner matrix.
ISSN:2331-8422