Riemann hypothesis from the Dedekind psi function

Abstract

Let $\mathcal{P}$ be the set of all primes and $\psi(n)=n\prod_{n\in \mathcal{P},p|n}(1+1/p)$ be the Dedekind psi function. We show that the Riemann hypothesis is satisfied if and only if $f(n)=\psi(n)/n-e^{\gamma} \log \log n <0$ for all integers $n>n_0=30$ (D), where $\gamma \approx 0.577$ is Euler's constant. This inequality is equivalent to Robin's inequality that is recovered from (D) by replacing $\psi(n)$ with the sum of divisor function $\sigma(n)\ge \psi(n)$ and the lower bound by $n_0=5040$. For a square free number, both arithmetical functions $\sigma$ and $\psi$ are the same. We also prove that any exception to (D) may only occur at a positive integer $n$ satisfying $\psi(m)/m<\psi(n)/n$, for any $m