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

7 pages, new proposition 3, many corrections

References in corpus (2)

• Remarks on Robin's and Niocolas Inequalities
• Le plus grand facteur premier de la fonction de Landau