Further results on generalized LCM matrices

Abstract

Let S={x ₁, x ₂, ∙∙∙, x _n } be a set of n distinct positive integers and f be an arithmetic function. By $\left( {\widehat f\left[ S \right]} \right)\left( {resp.\;\left( {\overline f \left[ S \right]} \right)} \right)$, we denote the n × n matrix whose i, j entry is $\sum\limits_{\mathop {\left[ {{x_i},{x_j}} \right]|l}\limits_{l \in S} } {f\left( l \right)} \;(resp.\sum\limits_{x \in S} {f\left( x \right)} - \sum\limits_{\mathop {{x_i}|l}\limits_{l \in S} } {f\left( l \right)} - \sum\limits_{\mathop {{x_j}|l}\limits_{l \in S} } {f\left( l \right)} + \sum\limits_{\mathop {\left[ {{x_i},{x_j}} \right]|l}\limits_{l \in S} } {f\left( l \right)} )$. In this paper, we first investigate the structures of the matrices $\left( {\widehat f\left[ S \right]} \right)$ and $\left( {\overline f \left[ S \right]} \right)$, then we give the formulae for the determinants of these matrices. These extend the results obtained by Bege in 2011. Finally, we give two examples to demonstrate the validity of our main results.