Note on some greatest common divisor matrices

• Autorzy: Peter Lindqvist , Kristian Seip
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11M41: Other Dirichlet series and zeta functions
• 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
• 11C20: Matrices, determinants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1998
• Tom: 84
• Numer: 2
• Strony: 149-154

Daty

wydano

1998

otrzymano

1997-05-09

Twórcy

autor

Peter Lindqvist

• Department of Mathematics, Norwegian Institute of Technology, N-7034 Trondheim, Norway

autor

Kristian Seip

• Department of Mathematics, Norwegian Institute of Technology, N-7034 Trondheim, Norway

Bibliografia

• [BL] K. Bourque and S. Ligh, On GCD and LCM matrices, Linear Algebra Appl. 174 (1992), 65-74.
• [C] S. Z. Chun, GCD and LCM power matrices, Fibonacci Quart. 34 (1996), 290-297.
• [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.
• [NZ] I. Niven and H. Zuckerman, An Introduction to the Theory of Numbers, 3rd ed., Wiley, New York, 1960.
• [S] H. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-6), 208-212.