A family of analogues to the Robin criterion

Abstract

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 $k\to\infty$. We prove the following analogue of the Robin criterion: for any $k \geq 2$, 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.