A family of analogues to the Robin criterion

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