On the Method for Proving the RH Using the Alcantara-Bode Equivalence

There is presented here a functional analysis - numerical solution for the Alcantara-Bode equivalent formulation of the Riemann Hypothesis (RH). The long-standing unsolved problem, posits that the non-trivial zeros of the Riemann Zeta function lie on the vertical line $\sigma=1/2$. Alcantara-Bode equivalent (1993) obtained from Beurling equivalent of RH (1955) states that RH holds if and only if the null space of a specific integral operator $T_\rho$ on $L^2(0,1)$ does not contain not null elements: $N_{T_\rho} = \{0\}$. The theory we introduced here is an update and extension of our previous work [1]. We provided methods for investigating the injectivity of linear bounded operators through their positivity properties on finite dimension subspaces. Here we separated the analysis of the operator restrictions from the approximations of the operator obtained by applying finite rank orthogonal projections. In both cases, the connection between the error estimations of an eligible zero and the positivity parameters dictates the operator injectivity. As a method, the Injectivity Criteria involving the adjoint operator introduced in [1] it is useful when no information we have to apply the finite rank approximation, like the compactness of the operator. The numerical evaluations using the finite rank operator approximations or operator restrictions on the same finite dimension subspaces, showed the injectivity of the integral operator from Alcantara-Bode equivalent. From Theorem 3 or Theorem 4 we obtained $ N_{T_\rho} = \{0\}$, meaning that half from Alcantara-Bode equivalent of RH holds. Then, the other half should hold, i.e.: the Riemann Hypothesis is true.