ON THE RIEMANN HYPOTHESIS AND THE DEDEKIND PSI FUNCTION

Let N_n = 2 · 3 · · · p_n be the primorial of order n and Ψ the Dedekind Psi function. Solé and Planat (2011) proved that the Riemann Hypothesis is true if and only if Ψ(N_n)/(N_n log log N_n) > eγ /ζ(2) for all n ≥ 3. We investigate the possibility of a reformulation of this criterion, where the term log N_n is replaced by the nth prime pn. Actually, we prove that if Ψ(N_n)/(N_n log p_n) > eγ /ζ(2) for all n ≥ 3, then the Riemann Hypothesis is true. Let ϕ denote the Euler totient function. As a consequence of the previous result, we obtain that if N_n/ϕ(N_n) > eγ log p_n for all n ≥ 3, then the Riemann Hypothesis is true