An upper bound for the \(\ell_p\) norm of a gcd-related matrix (Q2491548)

Let \(r\) and \(s\) be real numbers and consider the \(n\times n\) matrix \(M_{n}=[ m_{ij}] \) where \(m_{ij}:=(i,j)^{s}/[ i,j] ^{r}\) (here \((i,j)\) is the greatest common divisor (gcd) and \([ i,j] \) is the least common multiple). The author shows that, for any positive integer \(p\), \(r>1/p\) and \(s<r-1/p\), the values of \[ \| M_{n}\| _{p}:=\left( \sum_{i}\sum_{j}| m_{ij}| ^{p}\right) ^{1/p} \] are monotonic increasing with limit \(\zeta(rp)^{2/p}\zeta (rp-sp)^{1/p}\zeta(2rp)^{-1/p}\) as \(n\rightarrow\infty\) (\(\zeta\) is the Riemann zeta-function).