A Note on the Norms of the GCD Matrix

Let \(S=\{1,2,..., n\}\) be a set of positive integers. The \(n\times n\) matrix \([S]=(i,j)\), where \(s_{ij}=(x_{i},x_{j})\) the greatest common divisor of \(x_{i}\) and \(x_{j}\), is called the greatest common divisor GCD matrix on \(S\). In this study, we have obtained some bounds of norms of this matrix. In addition, we have obtained upper bounds of norms of the almost Hilbert-Smith GCD matrix is defined \((S)=\left[ \frac{(i,j)}{ij} \right]^{n}_{i,j=1}\)