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MORE ON A CERTAIN ARITHMETICAL DETERMINANT

Part of: Elementary number theory Polynomials and matrices Special matrices

Published online by Cambridge University Press:  17 October 2017

ZONGBING LIN  and
SIAO HONG
ZONGBING LIN
Affiliation:
Mathematical College, Sichuan University, Chengdu 610064, PR China School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, PR China email [email protected]
SIAO HONG*
Affiliation:
Center for Combinatorics, Nankai University, Tianjin 300071, PR China email [email protected]
*
✉[email protected]
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Abstract

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Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

Keywords

factor-closed setsquarefree integerarithmetical functionDirichlet convolutionSmith’s determinant

MSC classification

Primary: 11C20: Matrices, determinants
Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 15B36: Matrices of integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 15 - 25
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The research was partially supported partially by National Science Foundation of China Grant Nos. 11371260, 11771304 and 11671218.

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