Riemann hypothesis from the Dedekind psi function

Abstract

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