$\sigma$, $\varphi$, $\psi$ and $\zeta$

Let $\sigma(n)$ denote the sum of divisors of $n$. Robin showed that $\sigma(n)<\exp(\gamma)*n*\log(\log n)$ for every $n>5040$ if and only if the Riemann Hypothesis holds true (with $\gamma$ Euler's constant). Nicolas derived a similar, but rather easier to prove, criterion for the Euler totient function $\varphi$. We establish the analogue of his result for the Dedekind $\psi$-function, $\psi(n)=n \prod_{p|n}(1+1/p)$. As a result we can show that the Robin inequality holds for 7-th power free integers.  (Joint work with M. Planat.)