A family of analogues to the Robin criterion

The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $\sigma (n)<e^{\gamma }n\log \log n$ for all $n>5040$, where $\sigma (n)$ is the sum of divisors of $n$, and $\gamma$ is the Euler–Mascheroni constant. Define the family of functions $${\sigma }^{[k]}(n):=\sum _{[d_1,\dots ,d_k]=n}{d}_1\dots d_k$$ where $[d_1,\dots ,d_k]$ is the least common multiple of $d_1,\dots ,d_k$. These functions behave asymptotically like $\sigma (n{)}^k$ as $k\to \mathrm{\infty }$. We prove the following analogue of the Robin criterion: for any $k\ge 2$, the Riemann hypothesis holds if and only if ${\sigma }^{[k]}(n)<\frac{(e^{\gamma }n\log \log n{)}^k}{\zeta (k)}$ for all $n>2162160$, where $\zeta$ is the Riemann zeta function.