  In the framework of dimensional regularization, we propose a generalization
of the renormalization group equations in the case of the perturbative quantum
gravity that involves renormalization of the metric and of the higher order
Riemann curvature couplings. The case of zero cosmological constant is
considered. Solving the renormalization group (RG) equations we compute the
respective beta functions and derive the recurrence relations, valid at any
order in the Newton constant, that relate the higher pole terms $1/(d-4)^n$ to
a single pole $1/(d-4)$ in the quantum effective action. Using the recurrence
relations we find the exact form for the higher pole counter-terms that appear
in 2, 3 and 4 loops and we make certain statements about the general structure
of the higher pole counter-terms in any loop. We show that the complete set of
the UV divergent terms can be consistently (at any order in the Newton
constant) hidden in the bare gravitational action, that includes the terms of
higher order in the Riemann tensor, provided the metric and the higher
curvature couplings are renormalized according to the RG equations.
