论文信息 - Notes on the divisibility of GCD and LCM Matrices

Notes on the divisibility of GCD and LCM Matrices

Let S = { x 1 , x 2 , … , x n } be a set of positive integers, and let f be an arithmetical function. The matrices ( S ) f = [ f ( gcd ( x i , x j ) ) ] and [ S ] f = [ f ( lcm [ x i , x j ] ) ] are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices on S with respect to f , respectively. In this paper, we assume that the elements of the matrices ( S ) f and [ S ] f are integers and study the divisibility of GCD and LCM matrices and their unitary analogues in the ring M n ( ℤ ) of the n × n matrices over the integers.

Pentti Haukkanen | Ismo Korkee | P. Haukkanen | Ismo Korkee

[1] Pentti Haukkanen,et al. Some characterizations of totients , 1996 .

[2] Steve Ligh,et al. On GCD and LCM matrices , 1992 .

[3] Steve Ligh,et al. Matrices associated with multiplicative functions , 1995 .

[4] Pentti Haukkanen,et al. On meet and join matrices associated with incidence functions , 2003 .

[5] Shaofang Hong,et al. Factorization of matrices associated with classes of arithmetical functions , 2003 .

[6] Steve Ligh,et al. Matrices associated with arithmetical functions , 1993 .

[7] T. Apostol. Introduction to analytic number theory , 1976 .

[8] Pentti Haukkanen,et al. On Smith's determinant , 1997 .

[9] David Rearick. Semi-multiplicative functions , 1966 .

[10] P. McCarthy,et al. Introduction to Arithmetical Functions , 1985 .

[11] R Sivaramakrishnan,et al. Classical Theory of Arithmetic Functions , 1988 .

[12] Pentti Haukkanen,et al. On meet matrices on posets , 1996 .

[13] Shaofang Hong,et al. On the factorization of LCM matrices on gcd-closed sets☆ , 2002 .

[14] H. Smith,et al. On the Value of a Certain Arithmetical Determinant , 1875 .