Riemann hypothesis from the Dedekind psi function

TL;DR

This paper establishes a new criterion equivalent to the Riemann hypothesis involving the Dedekind psi function and demonstrates its relation to Robin's inequality, providing insights into prime number distribution.

Contribution

It introduces a novel inequality involving the Dedekind psi function that characterizes the truth of the Riemann hypothesis, linking it to classical divisor sum inequalities.

Findings

The Riemann hypothesis holds if and only if a specific inequality involving the Dedekind psi function is satisfied for all n > 30.

The inequality is equivalent to Robin's inequality when replacing the psi function with the sum of divisors function.

Exceptions to the inequality, if any, occur at integers with certain minimality properties related to the psi function ratio.

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