New results on the value of a certain arithmetical determinant

Let $m$ and $n$ be integers such that $1\le m\le n$. By \[G_{m,n}=(\gcd(i,j))_{m\le i,j\le n}\] we denote the $(n-m+1)\times(n-m+1)$ matrix having $\gcd(i,j)$ as its $i,j$-entry for all integers $i$ and $j$ between $m$ and $n$. Smith showed in 1875 that $\det(G_{1,n})=\prod\limits_{k=1}^n\varphi (k)$, where $\varphi $ is the Euler's totient function. In 2016, Hong, Hu and Lin proved that if $n\ge 2$ is an integer, then $\det(G_{2,n})=\Big(\prod\limits_{k= 1}^n\varphi(k)\Big)\sum\limits_{k=1\atop k \text{ is squarefree}}^n{\frac{1}{\varphi(k)}}$. In this paper, we show that if $n\ge 3$ is an integer, then $\det(G_{3,n})=\big(\sigma_0 \sigma_1+\frac{1}{2}\sigma_1\sigma_2+\frac{1}{2}\sigma_0\sigma_2\big)\prod\limits_{k= 1}^n\varphi(k)$, where for $i=0, 1$ and 2, one has $\sigma_i:=\sum\limits_{k=1\atop k \text{ is odd squarefree}}^{\lfloor\frac{n}{2^i}\rfloor}\frac{1}{\varphi(k)}$. Further, we calculate the determinants of the matrices $(f(\gcd(x_i,x_j)))_{1\le i,j\le n}$ and $(f({\rm lcm}(x_i,x_j)))_{1\le i,j\le n}$ having $f$ evaluated at $\gcd(x_{i}, x_{j})$ and ${\rm lcm}(x_{i},x_{j})$ as their $(i,j)$-entries, respectively, where $S=\{x_1,\ldots,x_n\}$ is a set of distinct positive integers such that $x_i>1$ for any integer $i$ with $1\le i\le n$, and $S\cup\{1,p\}$ is factor closed (that is, $S\cup\{1,p\}$ contains every divisor of $x$ for any $x\in S\cup\{1,p\}$), where $p\notin S$ is a prime number. Our result answers partially an open problem raised by Ligh in 1988.