  We present a detailed study of the total $p p$ cross section for scalar Higgs
production to next-to-next-to leading order in $\alpha_s$ at LHC energies, and
of the $p\bar p$ cross section at the Tevatron, combining an implementation of
the solutions of the parton evolution equations at the 3-loop order with the
corresponding hard scatterings, evaluated at the same perturbative order. Our
solutions of the DGLAP equations are implemented directly in $x$-space and
allow the study of the dependence of the results on the factorization ($\mu_F$)
and renormalization scales $(\mu_R)$ typical of a given process, together with
the stability of the perturbative expansion. The input sets for the parton
evolutions are those given by Martin, Roberts, Stirling and Thorne and by
Alekhin. Results for K-factors are also presented. The NNLO corrections can be
quite sizeable at typical collider energies. The stability region of the
perturbative expansion is found when $\mu_R > m_H\sim\mu_F$.
