Mathematics – Number Theory
Scientific paper
2008-08-26
Acta Arith. 134 (2008), 381-386.
Mathematics
Number Theory
8 pages. to appear in Acta Arith
Scientific paper
Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A positive semi-definite matrix $A=(a_{ij})$ with $a_{ij}\ge 0$ for all $i$ and $j$ is called infinitely divisible if the fractional Hadamard power $A^{\circ r}=(a_{ij}^r)$ is positive semi-definite for every nonnegative real number $r$. Let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive integers. In this paper, we show that if $f$ is a multiplicative function such that $(f*\mu)(d)\ge 0$ whenever $d|x$ for any $x\in S$, then the $n\times n$ matrices $(f(x_i, x_j))$, $(\frac{1}{f[x_i, x_j]})$ and $(\frac{f(x_i, x_j)}{f[x_i, x_j]})$ are infinitely divisible. Finally we extend these results to the Dirichlet convolution case which produces infinitely many examples of infinitely divisible matrices. Our results extend the results obtained previously by Bourque, Ligh, Bhatia, Hong, Lee, Lindqvist and Seip.
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