Research article

Divisibility among determinants of power matrices associated with integer-valued arithmetic functions

  • Received: 14 January 2020 Accepted: 13 February 2020 Published: 19 February 2020
  • MSC : 11C20, 11A05, 15B36

  • Let $a, b$ and $n$ be positive integers and $S = \left\{ {x_1, ..., x_n} \right\}$ be a set of $n$ distinct positive integers. The set $S$ is called a divisor chain if there is a permutation $\sigma $ of $\{1, ..., n\}$ such that $x_{\sigma (1)}|...|x_{\sigma (n)}$. We say that the set $S$ consists of two coprime divisor chains if we can partition $S$ as $S = S_1\cup S_2$, where $S_1$ and $S_2$ are divisor chains and each element of $S_1$ is coprime to each element of $S_2$. For any arithmetic function $f$, we define the function $f^a$ for any positive integer $x$ by $f^a(x): = (f(x))^a$. The matrix $(f^a(S))$ is the $n\times n$ matrix having $f^a$ evaluated at the the greatest common divisor of $x_{i}$ and $x_{j}$ as its $(i, j)$-entry and the matrix $(f^a[S])$ is the $n\times n$ matrix having $f^a$ evaluated at the least common multiple of $x_i$ and $x_j$ as its $(i, j)$-entry. In this paper, when $f$ is an integer-valued arithmetic function and $S$ consists of two coprime divisor chains with $1 \not\in S$, we establish the divisibility theorems between the determinants of the power matrices $(f^a(S))$ and $(f^b(S))$, between the determinants of the power matrices $(f^a[S])$ and $(f^b[S])$ and between the determinants of the power matrices $(f^a(S))$ and $(f^b[S])$. Our results extend Hong's theorem obtained in 2003 and the theorem of Tan, Lin and Liu gotten in 2011.

    Citation: Long Chen, Shaofang Hong. Divisibility among determinants of power matrices associated with integer-valued arithmetic functions[J]. AIMS Mathematics, 2020, 5(3): 1946-1959. doi: 10.3934/math.2020130

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  • Let $a, b$ and $n$ be positive integers and $S = \left\{ {x_1, ..., x_n} \right\}$ be a set of $n$ distinct positive integers. The set $S$ is called a divisor chain if there is a permutation $\sigma $ of $\{1, ..., n\}$ such that $x_{\sigma (1)}|...|x_{\sigma (n)}$. We say that the set $S$ consists of two coprime divisor chains if we can partition $S$ as $S = S_1\cup S_2$, where $S_1$ and $S_2$ are divisor chains and each element of $S_1$ is coprime to each element of $S_2$. For any arithmetic function $f$, we define the function $f^a$ for any positive integer $x$ by $f^a(x): = (f(x))^a$. The matrix $(f^a(S))$ is the $n\times n$ matrix having $f^a$ evaluated at the the greatest common divisor of $x_{i}$ and $x_{j}$ as its $(i, j)$-entry and the matrix $(f^a[S])$ is the $n\times n$ matrix having $f^a$ evaluated at the least common multiple of $x_i$ and $x_j$ as its $(i, j)$-entry. In this paper, when $f$ is an integer-valued arithmetic function and $S$ consists of two coprime divisor chains with $1 \not\in S$, we establish the divisibility theorems between the determinants of the power matrices $(f^a(S))$ and $(f^b(S))$, between the determinants of the power matrices $(f^a[S])$ and $(f^b[S])$ and between the determinants of the power matrices $(f^a(S))$ and $(f^b[S])$. Our results extend Hong's theorem obtained in 2003 and the theorem of Tan, Lin and Liu gotten in 2011.


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    [1] T. M. Apostol, Arithmetical properties of generalized Ramanujan Sums, Pacific J. Math., 41 (1972), 281-293. doi: 10.2140/pjm.1972.41.281
    [2] T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976.
    [3] S. Beslin and S. Ligh, Another generalisation of Smith's determinant, Bull. Austral. Math. Soc., 40 (1989), 413-415. doi: 10.1017/S0004972700017457
    [4] R. Bhatia, J. A. Dias da Silva, Infinite divisibility of GCD matrices, Amer. Math. Monthly, 115 (2008), 551-553. doi: 10.1080/00029890.2008.11920562
    [5] K. Bourque and S. Ligh, On GCD and LCM matrices, Linear Algebra Appl., 174 (1992), 65-74. doi: 10.1016/0024-3795(92)90042-9
    [6] K. Bourque and S. Ligh, Matrices associated with arithmetical functions, Linear Multilinear Algebra, 34 (1993), 261-267. doi: 10.1080/03081089308818225
    [7] K. Bourque and S. Ligh, Matrices associated with classes of arithmetical functions, J. Number Theory, 45 (1993), 367-376. doi: 10.1006/jnth.1993.1083
    [8] K. Bourque and S. Ligh, Matrices associated with classes of multiplicative functions, Linear Algebra Appl., 216 (1995), 267-275. doi: 10.1016/0024-3795(93)00154-R
    [9] W. Cao, On Hong's conjecture for power LCM matrices, Czechoslovak Math. J., 57 (2007), 253-268. doi: 10.1007/s10587-007-0059-3
    [10] P. Codeca and M. Nair, Calculating a determinant associated with multilplicative functions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (2002), 545-555.
    [11] W. D. Feng, S. F. Hong and J. R. Zhao, Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets, Discrete Math., 309 (2009), 2627-2639. doi: 10.1016/j.disc.2008.06.014
    [12] P. Haukkanen and I. Korkee, Notes on the divisibility of GCD and LCM matrices, Int. J. Math. Math. Sci., (2005), 925-935.
    [13] T. Hilberdink, Determinants of multiplicative Toeplitz matrices, Acta Arith., 125 (2006), 265-284. doi: 10.4064/aa125-3-4
    [14] S. A. Hong, S. N. Hu and Z. B. Lin, On a certain arithmetical determinant, Acta Math. Hungar., 150 (2016), 372-382. doi: 10.1007/s10474-016-0664-4
    [15] S. A. Hong and Z. B. Lin, New results on the value of a certain arithmetical determinant, Publicationes Mathematicae Debrecen, 93 (2018), 171-187. doi: 10.5486/PMD.2018.8139
    [16] S. F. Hong, On Bourque-Ligh conjecture of LCM matrices, Adv. Math. (China), 25 (1996), 566-568.
    [17] S. F. Hong, On the Bourque-Ligh conjecture of least common multiple matrices, J. Algebra, 218 (1999), 216-228. doi: 10.1006/jabr.1998.7844
    [18] S. F. Hong, Gcd-closed sets and determinants of matrices associated with arithmetical functions, Acta Arith., 101 (2002), 321-332. doi: 10.4064/aa101-4-2
    [19] S. F. Hong, On the factorization of LCM matrices on gcd-closed sets, Linear Algebra Appl., 345 (2002), 225-233. doi: 10.1016/S0024-3795(01)00499-2
    [20] S. F. Hong, Factorization of matrices associated with classes of arithmetical functions, Colloq. Math., 98 (2003), 113-123. doi: 10.4064/cm98-1-9
    [21] S. F. Hong, Nonsingularity of matrices associated with classes of arithmetical functions, J. Algebra, 281 (2004), 1-14. doi: 10.1016/j.jalgebra.2004.07.026
    [22] S. F. Hong, Divisibility properties of power GCD matrices and power LCM matrices, Linear Algebra Appl., 428 (2008), 1001-1008. doi: 10.1016/j.laa.2007.08.037
    [23] S. F. Hong and K. S. Enoch Lee, Asymptotic behavior of eigenvalues of reciprocal power LCM matrices, Glasgow Math. J., 50 (2008), 163-174. doi: 10.1017/S0017089507003953
    [24] S. F. Hong and R. Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasgow Math. J., 46 (2004), 551-569. doi: 10.1017/S0017089504001995
    [25] S. F. Hong and R. Loewy, Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r), Int. J. Number Theory, 7 (2011), 1681-1704. doi: 10.1142/S179304211100437X
    [26] S. F. Hong, J. R. Zhao and Y. Z. Yin, Divisibility properties of Smith matrices, Acta Arith., 132 (2008), 161-175. doi: 10.4064/aa132-2-4
    [27] I. Korkee and P. Haukkanen, On the divisibility of meet and join matrices, Linear Algebra Appl., 429 (2008), 1929-1943. doi: 10.1016/j.laa.2008.05.025
    [28] M. Li, Notes on Hong's conjectures of real number power LCM matrices, J. Algebra, 315 (2007), 654-664. doi: 10.1016/j.jalgebra.2007.05.005
    [29] M. Li and Q. R. Tan, Divisibility of matrices associated with multiplicative functions, Discrete Math., 311 (2011), 2276-2282. doi: 10.1016/j.disc.2011.07.015
    [30] Z. B. Lin and S. A. Hong, More on a certain arithmetical determinant, Bull. Aust. Math. Soc., 97 (2018), 15-25. doi: 10.1017/S0004972717000788
    [31] M. Mattila, On the eigenvalues of combined meet and join matrices, Linear Algebra Appl., 466 (2015), 1-20. doi: 10.1016/j.laa.2014.10.001
    [32] P. J. McCarthy, A generalization of Smith's determinant, Can. Math. Bull., 29 (1986), 109-113. doi: 10.4153/CMB-1986-020-1
    [33] H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc., 7 (1875), 208-213.
    [34] Q. R. Tan, Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains, Linear Multilinear Algebra, 58 (2010), 659-671. doi: 10.1080/03081080903008579
    [35] Q. R. Tan and M. Li, Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains, Linear Algebra Appl., 438 (2013), 1454-1466. doi: 10.1016/j.laa.2012.08.036
    [36] Q. R. Tan, Z. B. Lin and L. Liu, Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains II, Linear Multilinear Algebra, 59 (2011), 969-983. doi: 10.1080/03081087.2010.509721
    [37] Q. R. Tan, M. Luo and Z. B. Lin, Determinants and divisibility of power GCD and power LCM matrices on finitely many coprime divisor chains, Appl. Math. Comput., 219 (2013), 8112-8120.
    [38] J. X. Wan, S. N. Hu and Q. R. Tan, New results on nonsingular power LCM matrices, Electron. J. Linear Al., 27 (2014), 258.
    [39] A. Wintner, Diophantine approximations and Hilbert's space, Am. J. Math., 66 (1944), 564.
    [40] J. Xu and M. Li, Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains, Linear Multilinear Algebra, 59 (2011), 773-788. doi: 10.1080/03081087.2010.526942
    [41] Y. Yamasaki, Arithmetical properties of multiple Ramanujan sums, Ramanujan J., 21 (2010), 241-261. doi: 10.1007/s11139-010-9223-8
    [42] J. R. Zhao, Divisibility of power LCM matrices by power GCD matrices on gcd-closed sets, Linear Multilinear Algebra, 62 (2014), 735-748. doi: 10.1080/03081087.2013.786717
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