287
Views
5
CrossRef citations to date
0
Altmetric
Articles

Multiple divisor chains and determinants of matrices associated with completely even functions (mod r)

&
Pages 1240-1257 | Received 09 May 2013, Accepted 20 Jun 2013, Published online: 25 Aug 2013

References

  • Cohen E. Arithmetical inversion formulas. Canad. J. Math. 1960;12:399–409.
  • Smith HJS. On the value of a certain arithmetical determinant. Proc. London Math. Soc. 1875–1876;7:208–212.
  • Apostol TM. Arithmetical properties of generalized Ramanujan sums. Pacific J. Math. 1972;41:281–293.
  • McCarthy PJ. A generalization of Smith’s determinant. Canad. Math. Bull. 1986;29:109–113.
  • Cohen E. A class of arithmetical functions. Proc. Nat. Acad. Sci. U. S. A. 1955;41:939–944.
  • Bourque K, Ligh S. Matrices associated with classes of arithmetical functions. J. Number Theory. 1993;45:367–376.
  • Hong S. Gcd-closed sets and determinants of matrices associated with arithmetical functions. Acta Arith. 2002;101:321–332.
  • Codecá P, Nair M. Calculating a determinant associated with multiplicative functions. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 2002;5:545–555.
  • Hilberdink T. Determinants of multiplicative Toeplitz matrices. Acta Arith. 2006;125:265–284.
  • Hong S, Zhou X, Zhao J. Power GCD matrices for a UFD. Algebra Colloq. 2009;16:71–78.
  • Hong S. Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets. Linear Algebra Appl. 2006;416:124–134.
  • Hong S. Enoch Lee KS. Asymptotic behavior of eigenvalues of reciprocal power LCM matrices. Glasgow Math. J. 2008;50:163–174.
  • Hong S, Loewy R. Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r). Int. J. Number Theory. 2011;7:1681–1704.
  • Zhao J. Divisibility of power LCM matrices by power GCD matrices on GCD-closed sets. Linear Multilinear Algebra. Published online. 2013 Apr 30. doi:10.1080/03081087.2013.786717.
  • Hong S. Nonsingularity of matrices associated with classes of arithmetical functions. J. Algebra. 2004;281:1–14.
  • Hong S. Divisibility properties of power GCD matrices and power LCM matrices. Linear Algebra Appl. 2008;428:1001–1008.
  • Tan Q. Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains. Linear Multilinear Algebra. 2010;58:659–671.
  • Tan Q, Li M. Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains. Linear Algebra Appl. 2013;438:1454–1466.
  • Tan Q, Lin Z, Liu L. Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains II. Linear Multilinear Algebra. 2011;59:969–983.
  • Tan Q, Luo M, Lin Z. Determinants and divisibility of power GCD and power LCM matrices on finitely many coprime divisor chains. Appl. Math. Comput. 2013;219:8112–8120.
  • Xu J, Li M. Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains. Linear Multilinear Algebra. 2011;59:773–788.
  • Hong S, Zhao J, Yin Y. Divisibility properties of Smith matrices. Acta Arith. 2008;132:161–175.
  • Bourque K, Ligh S. Matrices associated with arithmetical functions. Linear Multilinear Algebra. 1993;34:261–267.
  • Hong S. Factorization of matrices associated with classes of arithmetical functions. Colloq. Math. 2003;98:113–123.
  • Apostol TM. Introduction to analytic number theory. New York: Springer-Verlag; 1976.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.