Riemann hypothesis from the Dedekind psi function

Let P be the set of all primes and ψ(n) = n ∏ n∈P,p|n(1 + 1/p) be the Dedekind psi function. We prove that, under the assumption that the n-th prime pn is always larger than the first Chebyshev function θ(pn), the Riemann hypothesis is satisfied if and only if f(n) = ψ(n)/n − e log logn < 0 for all integers n > n0 = 30 (D), where γ ≈ 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) ≥ ψ(n) and the lower bound by n0 = 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 < n, hence at a primorial number Nn or at one its multiples smaller than Nn+1 (Sloane sequence A060735). According to a Mertens theorem, all these candidate numbers are found to satisfy (D): this implies that the Riemann hypothesis is true. PACS numbers: 11A41, 11N37, 11M32