With this paper, a reformulation and improvement of the previous version v3 of [2] and a continuation of [1], we provided a solution for the Alcantara-Bode equivalent formulation ([3]) of the 165 years old Riemann Hypothesis - one of the Clay Inst. \textit{Millennium Problems}.
We consider it to be a solution in the areas of numerical analysis, applied math rather than in the field of number theory as expected but, for someone who needs to have an answer for RH they could find it here.
The Alcantara-Bode equivalence (1993) of the Riemann Hypothesis (RH, 1859) consists in the injectivity of a certain Hilbert-Schmidt integral operator, result obtained from Beurling equivalent formulation of RH (1955).
Both outstanding equivalences reduced RH to a problem dealing with the injectivity of a specific linear bounded operator on a separable Hilbert space.
The scenario behind Theorem 1, a generic result in the area of functional analysis and its associated methods is briefly described below. Given T a linear operator bounded on a separable Hilbert space H without zeros in a dense family of finite dimensional subspaces, its zeros, if any, should be in the difference set between the Hilbert space and the dense set of the union of the subspaces. The methods, the Corollary and the Injectivity Criteria exploit the strict positivity of the operator on the family subspaces as well as the density of the family in H, in order to obtain the sufficient conditions imposed to our operator for its injectivity.
With both methods we proved the Alcantara-Bode equivalence, meaning: the Riemann Hypothesis holds.