Note for the Riemann Hypothesis

Let Ψ(n) = n · ∏q n (1 + 1/q ) denote the Dedekind Ψ function where q n means the prime q divides n. Define, for n ≥ 3; the ratio R(n) = Ψ(n)/n · log log n where log is the natural logarithm. Let Nₙ = 2 · … · qₙ be the primorial of order n. A trustworthy proof for the Riemann hypothesis has been considered as the Holy Grail of Mathematics by several authors. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. There are several statements equivalent to the famous Riemann hypothesis. We prove if the inequality R(Nₙ₊₁) < R(Nₙ) holds for all primes qₙ (greater than some threshold), then the Riemann hypothesis is true. In this note, we show that the previous inequality always holds for all large enough prime numbers.