Note on some greatest common divisor matrices

Peter Lindqvist; Kristian Seip

Acta Arithmetica (1998)

  • Volume: 84, Issue: 2, page 149-154
  • ISSN: 0065-1036

Abstract

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Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.

How to cite

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Peter Lindqvist, and Kristian Seip. "Note on some greatest common divisor matrices." Acta Arithmetica 84.2 (1998): 149-154. <http://eudml.org/doc/207140>.

@article{PeterLindqvist1998,
abstract = {Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.},
author = {Peter Lindqvist, Kristian Seip},
journal = {Acta Arithmetica},
keywords = {greatest common divisor matrices; Smith determinant; Dirichlet series; Riesz basis; Euler totient function; asymptotic bounds for eigenvalues; Riemann zeta function},
language = {eng},
number = {2},
pages = {149-154},
title = {Note on some greatest common divisor matrices},
url = {http://eudml.org/doc/207140},
volume = {84},
year = {1998},
}

TY - JOUR
AU - Peter Lindqvist
AU - Kristian Seip
TI - Note on some greatest common divisor matrices
JO - Acta Arithmetica
PY - 1998
VL - 84
IS - 2
SP - 149
EP - 154
AB - Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.
LA - eng
KW - greatest common divisor matrices; Smith determinant; Dirichlet series; Riesz basis; Euler totient function; asymptotic bounds for eigenvalues; Riemann zeta function
UR - http://eudml.org/doc/207140
ER -

References

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  1. [BL] K. Bourque and S. Ligh, On GCD and LCM matrices, Linear Algebra Appl. 174 (1992), 65-74. 
  2. [C] S. Z. Chun, GCD and LCM power matrices, Fibonacci Quart. 34 (1996), 290-297. 
  3. [HLS] H. Hedenmalm, P. Lindqvist and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L²(0,1), Duke Math. J. 86 (1997), 1-37. Zbl0887.46008
  4. [NZ] I. Niven and H. Zuckerman, An Introduction to the Theory of Numbers, 3rd ed., Wiley, New York, 1960. Zbl0098.03602
  5. [S] H. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-6), 208-212. 

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