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 show if the inequality R(Nₙ₊₁) < R(Nₙ) holds for n big enough, then the Riemann hypothesis is true. In this note, we prove that R(Nₙ₊₁) < R(Nₙ) always holds for n big enough.